thèse de doctorat de mathématiques présentée à par...

Thèse de Doctorat de Mathématiques

présentée à

l’Université de Bretagne Occidentale

par

Aurélien Monteillet

Contribution à l’étude

d’équations de propagations de fronts

locales et non-locales

Thèse soutenue le 17 novembre 2008 devant le jury composé de :

Rapporteurs : Giovanni Bellettini Professeur à l’Université de Rome

Régis Monneau Ingénieur en chef des Ponts et ChausséesEcole Nationale des Ponts et ChausséesHabilité à diriger des recherches

Examinateurs : Guy Barles Professeur à l’Université de Tours

Cyril Imbert Maître de conférences à l’Université Paris DauphineHabilité à diriger des recherches

Marc Quincampoix Professeur à l’Université de Brest

Directeur de thèse : Pierre Cardaliaguet Professeur à l’Université de Brest

Remerciements

Bien sûr, mes premiers remerciements vont à Pierre Cardaliaguet, pour son en-cadrement durant ces années de thèse, pour sa disponibilité, sa gentillesse, son en-thousiasme. Il est pour beaucoup dans le plaisir que j’ai eu à passer ces années dethèse à Brest.

J’adresse mes sincères remerciements à Giovanni Bellettini et Régis Monneau quiont accepté la fastidieuse tâche de rapporter ce travail. Je remercie aussi chaleureuse-ment Guy Barles, Cyril Imbert et Marc Quincampoix d’avoir accepté de participerau jury.

Travailler avec Pierre Cardaliaguet, Guy Barles, Olivier Ley et Nicolas Forcadelfut un honneur et un plaisir. Je les en remercie. J’ai également beaucoup profitédes discussions avec Régis Monneau, de ses conseils et de ses idées. Au cours deces trois ans de thèse et de collaborations, j’ai souvent eu l’occasion de leur rendrevisite, à Tours ou à l’Ecole des Ponts et Chaussées. Merci à eux pour leur accueiltoujours chaleureux et personnel lors de ces visites.

A tous les membres de l’ACI “Mouvements d’interfaces avec termes non-locaux”et de l’ANR MICA, un grand merci pour les rencontres organisées, aussi fructueusesmathématiquement qu’humainement: Pierre Cardaliaguet, Guy Barles, Régis Mon-neau, Cyril Imbert, Olivier Ley et Nicolas Forcadel à nouveau, mais aussi An-tonin Chambolle, Christine Georgelin, Nathaël Alibaud, Ali Srour, Thierry TchambaTabet, Ahmad El Hajj, Sébastien Collin.

Je tiens aussi à exprimer ma gratitude à tous les membres du laboratoire et dudépartement de mathématiques de Brest pour leur accueil, et en particulier à ceuxavec lesquels j’ai collaboré dans le cadre de mon monitorat, pour leur confiance etleurs conseils.

Au cours de ma scolarité, j’ai eu la chance de rencontrer des professeurs demathématiques exceptionnels. Ils m’ont transmis leur passion, et ont très fortementinspiré ma vocation d’enseignant : je pense tout particulièrement à Hédi Chakrounet Frédéric Dupré, avec qui j’ai pu nouer par la suite des relations d’amitié aussiagréables que l’ont été nos relations de maître à élève. J’ai également beaucoupapprofondi mes connaissances auprès de Michel Quercia.

Je remercie mes amis romanais, lyonnais, rennais et brestois, heureusement tropnombreux pour être tous cités, pour tous les moments passés ensemble, studieux ou

iii

détendus.

Enfin un merci particulier à ma famille qui, dans des conditions de vie idéales,m’a toujours laissé libre de mon orientation, et a suivi tout mon parcours avec intérêtet fierté. Peut-être sont-ils même encore capables de réciter le titre du chapitre 1,qui depuis sa publication a fait le tour de Romans.

Et merci surtout à Cathy, grâce à qui cette famille s’agrandit.

iv

Publications issues de la thèse

Articles publiés ou acceptés

• Integral formulations of the geometric eikonal equation, paru dans Interfacesand Free Boundaries 9 (2007), 253-283.

• Minimizing movements for dislocation dynamics with a mean curvature term(avec N. Forcadel), paru dans ESAIM: Control, Optimisation and Calculus of Vari-ations 15 (2009), 214-244.

• Existence of weak solutions for general nonlocal and nonlinear second-orderparabolic equations (avec G. Barles, P. Cardaliaguet et O. Ley), paru dans NonlinearAnalysis: Theory, Methods and Applications 71 (2009), 2801–2810.

• Uniqueness results for nonlocal Hamilton-Jacobi equations (avec G. Barles,P. Cardaliaguet et O. Ley), paru dans Journal of Functional Analysis 257 (2009),1261–1287.

• Convergence of approximation schemes for nonlocal front propagation equa-tions, à paraître dans Mathematics of Computation.

v

Sommaire

Introduction générale 11 Rappels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1 Propagations de fronts . . . . . . . . . . . . . . . . . . . . . . 21.2 Solutions de viscosité des équations de Hamilton-Jacobi. . . . 51.3 Théorie géométrique de la mesure . . . . . . . . . . . . . . . . 9

1.3.1 Mesures de Hausdorff . . . . . . . . . . . . . . . . . 91.3.2 Fonctions BV et ensembles de périmètre fini . . . . . 9

2 Formulations intégrales de l’équation eikonale . . . . . . . . . . . . . 113 Equations non-locales sans principe de comparaison . . . . . . . . . . 14

3.1 Non respect du principe de comparaison . . . . . . . . . . . . 143.2 Deux modèles non-locaux . . . . . . . . . . . . . . . . . . . . 15

3.2.1 La dynamique des dislocations . . . . . . . . . . . . 153.2.2 Le système de FitzHugh-Nagumo . . . . . . . . . . . 16

3.3 Solutions faibles d’équations non-locales . . . . . . . . . . . . 174 Existence de solutions faibles pour des équations non-locales générales 185 Dynamique des dislocations avec terme de courbure moyenne . . . . . 206 Unicité de solutions faibles pour les deux problèmes modèle . . . . . . 227 Convergence de schémas d’approximation pour des équations non-

locales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Conclusion et Perspectives . . . . . . . . . . . . . . . . . . . . . . . . 28

Short introduction 31

1 Integral formulations of the geometric eikonal equation 451 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 Notation and tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.1 Viscosity solutions and set-theoretic approach . . . . . . . . . 482.2 Semiconvex functions and subdifferential of a convex function 512.3 BV functions and sets of finite perimeter . . . . . . . . . . . . 52

2.3.1 Functions of bounded variations . . . . . . . . . . . . 522.3.2 Sets of finite perimeter . . . . . . . . . . . . . . . . . 52

3 An integral formulation of the eikonal equation for subsolutions . . . 533.1 Equation satisfied by wθ in the viscosity sense . . . . . . . . . 543.2 Variational equation satisfied by wθ . . . . . . . . . . . . . . . 553.3 The integral formulation . . . . . . . . . . . . . . . . . . . . . 58

vii

SOMMAIRE

4 Conversely: from the integral formulation to the notion of subsolution 605 Regularity of the front . . . . . . . . . . . . . . . . . . . . . . . . . . 646 Corresponding results for supersolutions . . . . . . . . . . . . . . . . 75

6.1 The integral formulation for supersolutions . . . . . . . . . . . 756.2 The converse implication for supersolutions . . . . . . . . . . . 766.3 Regularity of the front . . . . . . . . . . . . . . . . . . . . . . 77

7 Global estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2 Existence of weak solutions for general nonlocal and nonlinearsecond-order parabolic equations 791 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802 Definition of weak solutions . . . . . . . . . . . . . . . . . . . . . . . 833 Assumptions on the Hamiltonians . . . . . . . . . . . . . . . . . . . . 844 Existence of weak solutions to (1.1) (unbounded case) . . . . . . . . . 85

4.1 The existence theorem . . . . . . . . . . . . . . . . . . . . . . 854.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.2.1 Dislocation dynamics equations . . . . . . . . . . . . 874.2.2 A FitzHugh-Nagumo type system . . . . . . . . . . . 88

5 Existence of weak solution to (1.1) (bounded case) . . . . . . . . . . . 925.1 The existence theorem . . . . . . . . . . . . . . . . . . . . . . 925.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3 Minimizing movements for dislocation dynamics with a mean cur-vature term 951 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962 Existence of minimizing movements . . . . . . . . . . . . . . . . . . . 102

2.1 F -minimizers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022.2 Minimizing movements . . . . . . . . . . . . . . . . . . . . . . 105

3 Regularity for F -minimizers . . . . . . . . . . . . . . . . . . . . . . . 1083.1 Existence of tangent cones . . . . . . . . . . . . . . . . . . . . 1093.2 The regularity results . . . . . . . . . . . . . . . . . . . . . . . 111

4 The upper and lower limits . . . . . . . . . . . . . . . . . . . . . . . . 1134.1 Velocity of E∗ and E∗ . . . . . . . . . . . . . . . . . . . . . . 1144.2 Regularity of E∗ and E∗ . . . . . . . . . . . . . . . . . . . . . 1154.3 Comparison at initial time . . . . . . . . . . . . . . . . . . . . 119

5 Minimizing movements and weak solutions . . . . . . . . . . . . . . . 1206 Comparison with the smooth flow . . . . . . . . . . . . . . . . . . . . 1217 Existence and uniqueness of a smooth solution . . . . . . . . . . . . . 123

7.1 Existence of smooth solutions for the local problem . . . . . . 1237.2 Existence of a smooth solution for the nonlocal problem . . . 126

4 Uniqueness results for nonlocal Hamilton-Jacobi equations 1331 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1342 Definition of weak solutions to (1.1) . . . . . . . . . . . . . . . . . . . 1373 Model problem 1: dislocation type equations . . . . . . . . . . . . . . 137

viii

SOMMAIRE

4 Model problem 2: a FitzHugh-Nagumo type system . . . . . . . . . . 1404.1 Classical estimates for the inhomogeneous heat equation. . . . 1414.2 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . 1424.3 Proof of Lemma 4.4 . . . . . . . . . . . . . . . . . . . . . . . . 144

5 Eikonal equation, interior cone property and perimeter estimates . . . 1465.1 Some results on the classical eikonal equation . . . . . . . . . 1465.2 Estimates on the measure of level-sets for solutions of (1.3). . 1475.3 Estimate of the perimeter of sets with the interior cone property1485.4 Propagation of the interior cone property . . . . . . . . . . . . 152

5 Convergence of approximation schemes for nonlocal front propaga-tion equations 1591 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1602 Convergence of approximation schemes . . . . . . . . . . . . . . . . . 1633 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

3.1 Dislocation dynamics . . . . . . . . . . . . . . . . . . . . . . . 1703.2 A FitzHugh-Nagumo type system . . . . . . . . . . . . . . . . 174

ix

Introduction générale

L’objet des travaux exposés dans ce mémoire est la compréhension de certainsproblèmes liés aux propagations de fronts. Le cadre de travail est celui dessolutions de viscosité, qui permettent de définir des évolutions non régulières, dansles cas très courants où il n’existe pas d’évolution classique.

Le chapitre 1 porte sur l’étude de certaines propriétés de régularité des solutionsde la plus classique des équations géométriques locales, l’équation eikonale. Ons’intéresse notamment à des estimations sur leur périmètre, en particulier dans lecas où la vitesse peut changer de signe.

Le reste – la plus grande partie – de cette thèse est consacré aux équations non-locales, pour lesquelles la difficulté majeure est l’absence de principe d’inclusion,c’est-à-dire que deux fronts bien ordonnés au temps initial peuvent être amenés à se“croiser”. Les travaux présentés ont pour objet à la fois l’étude générale de ces équa-tions, et la compréhension plus particulière de deux modèles riches en difficultés, ladynamique des dislocations et un système de type FitzHugh-Nagumo. A cause del’absence de principe d’inclusion, à la fois l’existence et l’unicité de solutions de vis-cosité classiques sont des questions ouvertes en général. Pour pallier cette difficulté,on s’intéresse à une notion de solution faible pour ces équations. Dans le chapitre2, on établit un résultat d’existence général de telles solutions, qui est appliqué auxdeux modèles. Le chapitre 3 est consacré à l’étude d’un mode de la dynamique d’unedislocation avec un terme additionnel de courbure moyenne. L’aspect régularisantde ce terme nous permet de construire, par la méthode des mouvements minimisants,des solutions faibles particulières, plus proches d’une solution de viscosité classiqueque celles fournies par les résultats d’existence du chapitre 2. Dans le chapitre4, on prouve l’unicité de solution faible pour les deux équations modèle, dans lecas de vitesses positives. Enfin dans le chapitre 5, on s’intéresse à l’approximationnumérique de ces propagations non-locales, et notamment à la convergence de sché-mas abstraits vers une solution faible. Des exemples explicites de tels schémas sontaussi analysés pour les deux modèles.

Commençons par une présentation générale des notions utilisées, des difficultésqui se posent et des résultats que nous avons obtenus.


1 Rappels

1.1 Propagations de fronts

On entend par front le bord Γ d’un fermé K ⊂ RN (N ∈ N∗), et l’on parlede propagation lorsque ce front dépend du temps. On note alors Γ(t) = ∂K(t) lefront au temps t, pour t ∈ [0, T ] où T > 0 est un temps final. Dans le cas d’uneévolution régulière, le type de lois de propagation auxquelles on s’intéresse est celuioù la vitesse normale Vx,t de Γ(t) au point x est fonction:

– du point x et du temps t,– de la normale extérieure unitaire νx,t à K(t) au point x,– de la matrice courbure Ax,t = [− ∂νi

∂xj(x, t)]1≤i,j≤N de K(t) au point x (négative

pour les convexes),– éventuellement, de la famille K(s)s∈[0,T ]. Dans ce cas, l’équation est dite

non-locale.

νx,t

x •K(t)

Γ(t)

Figure 1: Propagation de front dans la direction normale.

On écrira donc la loi d’évolution de la façon suivante:

Vx,t = h(x, t, νx,t, Ax,t), (1.1)

ou, dans le cas d’une équation non-locale,

Vx,t = h[K](x, t, νx,t, Ax,t), (1.2)

où h (resp. h[K]) est une fonction définie sur RN × [0, T ] × SN−1 × SN , la notationSN−1 désignant la sphère unité de RN , et SN désignant l’ensemble des matricescarrées symétriques réelles de taille N .

Rappelons, dans le cas d’une évolution régulière, la notion de vitesse normale deΓ(t) en un point x: pour δ assez petit, soit

γ : ]t− δ, t+ δ[ 7→ RN

une application de classe C1 telle que pour tout s ∈ ]t− δ, t+ δ[, γ(s) ∈ Γ(s), et telleque γ(t) = x. Alors par définition, la vitesse normale de Γ(t) en x est donnée par leproduit scalaire

Vx,t = 〈νx,t, γ(t)〉.

2

1. Rappels

Méthode level-set

Avec cette définition, les équations géométriques (1.1)–(1.2) sont difficiles à manip-uler. De plus les définitions mêmes de normale, et par suite de vitesse normale, oucelle de courbure, n’ont pas de sens si Γ(t) n’est pas régulier, comme c’est le casen l’un des points de la Figure 1. Or, des singularités peuvent apparaître en tempsfini même lorsque le front initial est régulier: on peut penser par exemple à deuxdisques initialement disjoints évoluant à vitesse 1. Il est pourtant naturel de vouloiraussi propager de tels fronts après l’apparition de singularités. La méthode level-setde Sethian et Osher [64] permet d’adopter un point de vue bien plus fructueux surces problèmes. Pour la présenter, supposons que le front Γ(t) = ∂K(t) est régulier,et qu’il existe une application u : RN × [0, T ] → R régulière telle que

K(t) = x ∈ RN ; u(x, t) ≥ 0, Γ(t) = x ∈ R

N ; u(x, t) = 0,

et telle que le gradient de u en espace vérifie Du(x, t) 6= 0 si x ∈ Γ(t).Pour un tel x on sait que

νx,t = − Du(x, t)

|Du(x, t)| ,

où | · | désigne la norme euclidienne sur RN , tandis que

Ax,t =1

|Du(x, t)|

(IN − Du(x, t) ⊗Du(x, t)

|Du(x, t)|2)D2u(x, t),

où D2u(x, t) désigne la matrice Hessienne de u par rapport à la variable d’espace en(x, t), IN la matrice identité de taille N , et ⊗ le produit tensoriel entre vecteurs deR

N .

Calculons Vx,t; soit γ : ]t− δ, t+ δ[ 7→ RN une application de classe C1 telle quepour tout s ∈ ]t− δ, t+ δ[, γ(s) ∈ Γ(s), et telle que γ(t) = x. Par définition de u etγ, on a pour tout s ∈ ]t− δ, t+ δ[,

u(γ(s), s) = 0.

En dérivant cette expression par rapport à s, on obtient donc en s = t:

〈Du(x, t), γ(t)〉 + ut(x, t) = 0,

où ut désigne la dérivée partielle de u par rapport à la variable de temps t. On adonc

ut(x, t) = 〈−Du(x, t), γ(t)〉 = 〈νx,t, γ(t)〉 |Du(x, t)| = Vx,t |Du(x, t)|.

Ceci est valable a priori uniquement pour tout x tel que u(x, t) = 0, c’est-à-direpour la ligne de niveau 0 de u, mais en faisant évoluer toutes les lignes de niveau deu selon la même loi, on obtient finalement que u vérifie l’équation suivante, appeléeéquation level-set associée à (1.2), dans tout RN × [0, T ]:

ut = h[K]

(x, t,− Du

|Du| ,1

|Du|

(IN − Du⊗Du

|Du|2)D2u

)|Du|.

3


Pour certaines équations non-locales, qui sont celles auxquelles on s’intéressera, ilest possible de transposer le problème entièrement en termes de fonctions et non desdonnées géométriques telles que K, en écrivant l’équation précédente sous la forme

ut(x, t) = H [1u≥0](x, t,Du(x, t), D2u(x, t)), (1.3)

où 1A désigne la fonction indicatrice d’un ensemble A:

1A(x) =

1 si x ∈ A,

0 sinon.

Dans l’équation (1.3), le Hamiltonien H [χ] est défini pour tout χ ∈ L∞(RN × [0, T ])tel que 0 ≤ χ ≤ 1, et lorsque χ est une fonction indicatrice, il vérifie

H [χ](x, t, p, A) = h[χ = 1](x, t,− p

|p| ,1

|p|

(IN − p⊗ p

|p|2)A

)|p|

pour tout (x, t, p, A) ∈ RN × [0, T ] × RN \ 0 × SN .

Lorsque l’ensemble Γ(t) = u(·, t) = 0 n’est pas régulier en un point x, alors,essentiellement, soit Du(x, t) n’est pas défini, soit Du(x, t) = 0. Si l’on est capablede donner un sens à (1.3) dans ces deux cas, on peut alors définir un front généralisémême après l’apparition de singularités. Cette question est l’objet de la sectionsuivante.

Pour les équations dites locales, la dépendance de H par rapport à u ≥ 0n’existe pas. C’est le cas dans les deux exemples fondamentaux suivants:

Exemple 1.1 (Equations locales).

1. L’équation eikonale Vx,t = c(x, t). L’équation level-set associée est

ut(x, t) = c(x, t)|Du(x, t)|.

2. Le mouvement par courbure moyenne Vx,t = Tr(Ax,t). L’équation level-setassociée est

ut(x, t) = div

(Du

|Du|

)(x, t) |Du(x, t)| = ∆u(x, t)−〈D2u(x, t)Du(x, t), Du(x, t)〉

|Du(x, t)|2 .

Deux exemples d’équations non-locales, que nous étudierons en détails par la suite,seront présentés dans la section 3.2.

Dans tous les cas, la méthode level-set consiste en la réalisation des trois pointssuivants, afin de définir un front généralisé aussi lorsque Γ(0) n’est pas régulier, ouaprès l’apparition de singularités:

1. Trouver u0 : RN → R tel que

K(0) = x ∈ RN ; u0(x) ≥ 0 et Γ(0) = x ∈ R

N ; u0(x) = 0.

4

1. Rappels

2. Résoudre en un sens à préciser le problèmeut(x, t) = H [1u≥0](x, t,Du(x, t), D

2u(x, t)) pour (x, t) ∈ RN× ]0, T [,

u(x, 0) = u0(x) pour x ∈ RN .

(1.4)

3. Poser

K(t) = x ∈ RN ; u(x, t) ≥ 0 et Γ(t) = x ∈ R

N ; u(x, t) = 0.

On répond facilement à la première question: il suffit de choisir pour u0 la distancesignée à Γ(0) définie par

d(x) =

−dist(x,K(0)) pour x ∈ RN \K(0)

dist(x,Γ(0)) pour x ∈ K(0).

On peut donc même, sans perte de généralité, supposer que u0 est Lipschitzienne.

Pour les équations non-locales, la notation H [1u≥0] est justifiée par le fait quepour chaque fonction indicatrice 1u≥0, ou plus généralement pour chaque fonctionχ ∈ L∞(RN × [0, T ]) fixée vérifiant 0 ≤ χ ≤ 1, H [χ] définit un Hamiltonien pourlequel on sait résoudre l’équation

ut(x, t) = H [χ](x, t,Du(x, t), D2u(x, t)), (1.5)

qui est une équation locale. La résolution de (1.4) correspond alors au cas où χest lié à la solution u de (1.5) avec donnée initiale u0 par la relation χ = 1u≥0.La réponse à la seconde question passe donc par la résolution des équations localesdu type (1.5); elle est donnée par la notion de solution de viscosité de Crandall etLions [37]. Pour des présentations complètes de cette théorie, on pourra consulternotamment [36, 14], ainsi que [46] pour la théorie spécifique aux fronts. Donnonsseulement ici leur définition et quelques unes de leurs principales propriétés.

1.2 Solutions de viscosité des équations de Hamilton-Jacobi.

Soit T > 0 et H une application continue de RN × [0, T ] × RN \ 0 × SN dans R.On considère l’équation

ut = H(x, t,Du,D2u) dans RN× ]0, T [. (1.6)

On supposera toujours que cette équation est parabolique, c’est-à-dire que H estcroissante par rapport à la variable A:

H(x, t, p, A) ≤ H(x, t, p, B) si A ≤ B,

la relation A ≤ B étant comprise au sens des matrices symétriques. PuisqueH(x, t, p, A) n’est pas nécessairement défini en p = 0 (cf exemple 1.1, 2.), on vacommencer par étendre H à RN × [0, T ] × RN × SN de deux façons:

H∗(x, t, 0, A) = lim sup(xn,tn,pn,An)→(x,t,0,A); pn 6=0

H(xn, tn, pn, An),

5


H∗(x, t, 0, A) = lim inf(xn,tn,pn,An)→(x,t,0,A); pn 6=0

H(xn, tn, pn, An).

On introduit aussi les enveloppes semi-continue supérieure et inférieure d’une fonc-tion localement bornée u : R

N × [0, T ] → R par

u∗(x, t) = lim sup(xn,tn)→(x,t)

u(xn, tn),

u∗(x, t) = lim inf(xn,tn)→(x,t)

u(xn, tn).

Définition 1.2 (Solutions de viscosité).1. On dit que u : RN × [0, T ] → R est sous-solution de viscosité de

ut = H(x, t,Du,D2u)

dans RN× ]0, T [ si pour toute fonction test φ ∈ C2(RN× ]0, T [; R) telle que u∗− φ aun maximum local en un certain (x0, t0) ∈ RN× ]0, T [, on a

φt(x0, t0) ≤ H∗(x0, t0, Dφ(x0, t0), D2φ(x0, t0)).

2. On dit que u est sur-solution de viscosité de ut = H(x, t,Du,D2u) dansRN× ]0, T [ si pour toute fonction test φ ∈ C2(RN× ]0, T [; R) telle que u∗ − φ a unminimum local en un certain (x0, t0) ∈ RN× ]0, T [, on a

φt(x0, t0) ≥ H∗(x0, t0, Dφ(x0, t0), D2φ(x0, t0)).

3. Enfin on dit que u est solution de viscosité de ut = H(x, t,Du,D2u) dansRN× ]0, T [ si u en est à la fois sous-solution et sur-solution.

Dans le cas d’un problème de Cauchyut = H(x, t,Du,D2u) dans RN×]0, T [,

u(x, 0) = u0 dans RN ,

où u0 : RN → R est une fonction continue, on rajoutera de plus dans cette définition

la condition u∗(·, 0) ≤ u0 pour une sous-solution, et la condition u∗(·, 0) ≥ u0 pourune sur-solution.

Propriétés fondamentales des solutions de viscosité

Principe de comparaison

Un outil fondamental, notamment pour l’unicité de solution, est le principe de com-paraison:

Définition 1.3 (Principe de comparaison).On dit que l’équation ut = H(x, t,Du,D2u) satisfait un principe de comparaison

si pour toute sous-solution de viscosité u de cette équation, et toute sur-solution deviscosité v,

u∗(x, 0) ≤ v∗(x, 0) dans RN ⇒ u∗(x, t) ≤ v∗(x, t) dans R

N × [0, T [.

6

1. Rappels

On pourra consulter notamment [36, 22, 14, 46] pour des conditions sur H souslesquelles l’équation ut = H(x, t,Du,D2u) satisfait un principe de comparaison.

Géométriquement, le principe de comparaison correspond à un principe d’inclusionsur les ensembles K(t) = u(·, t) ≥ 0 et K ′(t) = v(·, t) ≥ 0:

K(0) ⊂ K ′(0) ⇒ K(t) ⊂ K ′(t) pour tout t ∈ [0, T [.

Ce principe n’est pas toujours vérifié pour les équations non-locales. C’est la sourcemajeure des difficultés liées à ces équations auxquelles une grande partie de cettethèse est consacrée. Nous y reviendrons dès la section 3.

Stabilité

La notion de solution de viscosité est très souple vis-à-vis des passages à la lim-ite, plus précisément à la “semi-limite relaxée”, une notion introduite par Barles etPerthame [21]. Soit (un) une suite localement majorée (resp. minorée) d’applicationsdéfinies sur RN × [0, T ] à valeurs réelles. On appelle semi-limite relaxée supérieure(resp. inférieure) de la suite (un) l’application lim sup∗(un) (resp. lim inf∗(un))définie par

lim sup∗(un)(x, t) = lim sup un(xn, tn), n→ +∞, (xn, tn) → (x, t)(resp. lim inf∗(un)(x, t) = lim inf un(xn, tn), n→ +∞, (xn, tn) → (x, t) ).

On a alors le théorème de stabilité suivant:

Théorème 1.4 (Stabilité, voir [21]).Si (un) est une suite localement majorée (resp. minorée) de sous-solutions de

viscosité (resp. sur-solutions) de ut = Hn(x, t,Du,D2u) dans RN× ]0, T [, et si (Hn)converge localement uniformément vers H, alors u∗ (resp. u∗) est sous-solution deviscosité (resp. sur-solution) de ut = H(x, t,Du,D2u) dans RN× ]0, T [.

Dans le cadre des équations non-locales, nous aurons besoin d’un tel résultat destabilité prenant en compte des convergences plus faibles des Hamiltoniens.

7


Invariance

Toutes les équations du type (1.6) obtenues par la méthode level-set sont ce qu’onappelle géométriques, c’est-à-dire que pour tout (x, t, p, A) et tous λ > 0, µ ∈ R,

H(x, t, λ p, λA+ µ p⊗ p) = λH(x, t, p, A).

Pour ces équations, on a un résultat remarquable appelé théorème d’invariance:

Théorème 1.5 (Invariance, voir [46]).Si l’équation ut = H(x, t,Du,D2u) est géométrique, et si u en est une sous-

solution de viscosité (resp. sur-solution) dans RN× ]0, T [, alors pour tout θ : R → R

continu croissant, θ u en est aussi sous-solution de viscosité (resp. sur-solution).

Ce résultat permet de construire d’autres solutions à partir d’une solution con-nue, et est utilisé par exemple pour montrer que:

– dans la méthode level-set, le front ne dépend pas de la fonction auxiliaire u0

choisie,– si une évolution géométrique régulière existe (i.e. un front régulier Γ évoluant

avec la vitesse donnée par (1.1)), elle coïncide avec la ligne de niveau 0 detoute solution de viscosité de l’équation level-set (1.6) correspondante.

Solutions de viscosité L1

Comme on le verra dans la section 3.3, nous aurons besoin d’une notion desolution de viscosité pour des Hamiltoniens ayant une dépendance en temps seule-ment mesurable. Il s’agit de la notion de solution de viscosité L1 pour laquelle onpourra consulter [50, 61, 62, 27, 26]. Nous donnons seulement ici la définition desous-solution de viscosité L1, les notions de sur-solution et solution étant définies defaçon symétrique comme dans la Définition 1.2.

Définition 1.6. Soit H : (x, t, p, A) ∈ RN×[0, T ]×RN\0×SN 7→ H(x, t, p, A) uneapplication continue par rapport à (x, p, A) pour presque tout t ∈ [0, T ], et mesurablesur [0, T ] pour tout (x, p, A).

On dit que u : RN × [0, T ] → R est sous-solution de viscosité L1 de

ut = H(x, t,Du,D2u)

dans RN× ]0, T [ si

(i) pour toute fonction φ ∈ C2(RN× ]0, T [; R) et toute fonction b ∈ L1(]0, T [; R) telleque u∗ − φ−

∫ t

0b(s) ds a un maximum local en un certain (x0, t0) ∈ RN× ]0, T [,

(ii) et pour toute fonction continue G : RN × [0, T ] × RN × SN → R telle que

H∗(x, t, p, A) − b(t) ≤ G(x, t, p, A)

pour presque tout t dans un voisinage de t0, et tout (x, p, A) dans un voisinage de(x0, Dφ(x0, t0), D

2φ(x0, t0)),

on aφt(x0, t0) ≤ G(x0, t0, Dφ(x0, t0), D

2φ(x0, t0)).

8

1. Rappels

Les propriétés classiques des solutions de viscosité restent valides dans ce cadre,comme le montrent les références citées plus haut.

Avec ces outils, nous sommes capables de définir des propagations de frontsgénéralisées pour les lois géométriques (1.1). Comme nous le verrons plus bas,une des questions fondamentales dans l’étude de ces propagations de fronts est lacompréhension de leur régularité, à commencer par la définition même de cette régu-larité: nous travaillons maintenant avec des fronts non lisses en général. La théoriegéométrique de la mesure, que nous présentons maintenant, consiste en l’étude despropriétés fines de géométrie différentielle des ensembles non réguliers de RN . C’estl’outil qui permet de comprendre les propriétés de “régularité” que possède le frontΓ(t), qui sont une préoccupation constante de cette thèse.

1.3 Théorie géométrique de la mesure

Nous allons donc introduire quelques concepts de théorie géométrique de lamesure: les mesures de Hausdorff de dimension s sur R

N , la notion d’ensemble depérimètre fini. On pourra notamment consulter [38] pour une présentation détailléece cette théorie.

1.3.1 Mesures de Hausdorff

Pour tous A ⊂ RN , s ≥ 0 et δ > 0, on définit

Hsδ(A) = inf

+∞∑

j=1

ωs

[diam(Cj)

2

]s

; A ⊂⋃

j≥1

Cj, diam(Cj) ≤ δ

,

où ωs est le volume de la boule unité de Rs lorsque s est un entier non nul. Ondéfinit aussi

Hs(A) = limδ→0

Hsδ(A).

Hs est appelée mesure de Hausdorff de dimension s sur RN . La mesure HN coincideavec la mesure de Lebesgue LN . L’exemple qui nous intéressera plus particulièrementest celui de la mesure “surfacique” HN−1.

1.3.2 Fonctions BV et ensembles de périmètre fini

Soit Ω un ouvert de RN .

Définition 1.7. Une application f ∈ L1loc(Ω) est dite à variations localement bornées

dans Ω si pour tout ouvert U ⋐ Ω (i.e. borné et vérifiant U ⊂ Ω),

sup

∫

U

f(x) div φ(x) dx ; φ ∈ C1c (U ; RN) ; ‖φ‖∞ ≤ 1

< +∞.

On note BVloc(Ω) l’ensemble des fonctions à variations localement bornées dans Ω.On dit de même que f ∈ L1(Ω) est à variations bornées dans Ω si la définition

précédente est valable pour U = Ω. On note BV (Ω) leur ensemble.

9


Par exemple, lorsque f est de classe C1 sur un ouvert Ω, alors pour tout ou-vert U ⋐ Ω et pour tout φ ∈ C1

c (U ; RN) tel que ‖φ‖∞ ≤ 1, on a par la formuled’intégration par parties,

∫

U

f(x) div φ(x) dx = −∫

U

〈Df(x), φ(x)〉 dx,

de sorte que f est à variations localement bornées dans Ω avec

sup

∫

U

f(x) div φ(x) dx ; φ ∈ C1c (U ; RN ) ; ‖φ‖∞ ≤ 1

= ‖Df‖L1(U).

En fait, cette situation se généralise aux fonctions à variations bornées. Le théorèmede représentation de Riesz donne en effet le résultat suivant:

Théorème 1.8. Soit f ∈ BVloc(Ω). Alors il existe une mesure de Radon µ sur Ωet une application µ-mesurable σ : Ω → R

N telle que:

1. |σ(x)| = 1 µ-presque partout.

2.∫Ωf(x) div φ(x) dx = −

∫Ω〈φ(x), σ(x)〉 dµ pour tout φ ∈ C1

c (Ω; RN).

La mesure µ est appelée mesure de variation de f , notée ‖Df‖. On a de plus pourtout ouvert U ⋐ Ω,

sup

∫

U

f(x) div φ(x) dx ; φ ∈ C1c (U ; RN ) ; ‖φ‖∞ ≤ 1

= ‖Df‖(U).

La notion de fonction à variations bornées permet de définir celle d’ensemble depérimètre fini:

Définition 1.9. Un ensemble LN -mesurable E ⊂ RN est dit de périmètre (locale-ment) fini dans Ω si 1E est à variations (localement) bornées dans Ω. Si Ω n’est pasmentionné, on sous-entend que Ω = R

N .

La mesure de variation de 1E est alors notée ‖∂E‖, et la fonction −σ donnéepar le Théorème 1.8 est notée νE. On a donc pour tout φ ∈ C1

c (Ω; RN),∫

E

div φ(x) dx =

∫

Ω

〈φ(x), νE(x)〉 d‖∂E‖.

Expliquons pourquoi ceci est la bonne notion de périmètre. Imaginons quel’ensemble E soit un ouvert borné de classe C2, et notons nE sa fonction normaleunitaire extérieure. Alors pour tout ouvert U ⋐ Ω et pour tout φ ∈ C1

c (U ; RN ) telque ‖φ‖∞ ≤ 1, on a par la formule de Gauss-Green,

∫

E

div φ(x) dx =

∫

∂E

〈φ(x), nE(x)〉 dHN−1,

de sorte que

sup

∫

E

div φ(x) dx ; φ ∈ C1c (U ; RN ) ; ‖φ‖∞ ≤ 1

= HN−1(∂E ∩ U).

On perçoit donc l’analogie avec la Définition 1.9, et on a l’intuition que:

10

2. Formulations intégrales de l’équation eikonale

1. νE est une fonction normale unitaire extérieure.

2. ‖∂E‖ = HN−1⌊∂E ,

où HN−1⌊A , pour A ⊂ R

N , désigne la restriction de HN−1 à l’ensemble A. C’est bienle cas, à condition de définir la bonne notion de bord ∂E:

Définition 1.10. Soit E un ensemble de périmètre localement fini dans Ω. On ditque x ∈ Ω appartient au bord réduit de E, noté ∂∗E, si:

1. ‖∂E‖(B(x, r)) > 0 pour tout r > 0 tel que B(x, r) ⊂ Ω,

2. 1‖∂E‖(B(x,r))

∫B(x,r)

νE d‖∂E‖ −→r→0

νE(x),

3. |νE(x)| = 1.

On a alors le résultat suivant:

Théorème 1.11. Soit E un ensemble de périmètre localement fini dans Ω. Alors:

1. ‖∂E‖(RN \ ∂∗E) = 0.

2. ‖∂E‖(U) = HN−1(∂∗E ∩ U) pour tout borélien U ⊂ Ω.

On appelle cette quantité le périmètre de E dans U , et on la note P (E,U). LorsqueE est de périmètre fini, on note aussi P (E) = P (E,RN), appelé périmètre de E.

3. Formule de Gauss-Green. Pour tout φ ∈ C1c (Ω; RN),

∫

E

div φ(x) dx =

∫

∂∗E

〈φ(x), νE(x)〉 dHN−1(x).

Par analogie avec l’exemple d’une fonction de classe C1, on a en quelque sorte(de façon très formelle), si E est de périmètre fini et si φ ∈ C1

c (RN ; RN),

∫

E

div φ(x) dx =

∫

RN

1E(x) div φ(x) dx “ = ” −∫

RN

〈D1E(x), φ(x)〉 dx

=

∫

∂∗E

〈φ(x), νE(x)〉 dHN−1(x),

et doncD1E × LN = −νE × HN−1

⌊∂∗E. (1.7)

Le cadre de travail étant posé, nous sommes maintenant en mesure d’exposer lestravaux contenus dans ce mémoire.

2 Formulations intégrales de l’équation eikonale

Commençons par présenter les résultats obtenus dans le chapitre 1 sur l’équationeikonale. Le but de ce travail est l’étude de la régularité des solutions de l’équationeikonale Vx,t = c(x, t), et plus précisément de leur périmètre. Outre un intérêt in-dépendant, il est utile dans certains problèmes de propagations de fronts non-localesde comprendre le comportement des solutions de l’équation eikonale, car celle-ci

11


apparaît naturellement en “gelant” le terme non-local, typiquement dans les tech-niques de point fixe: c’est ce que suggère (1.5). Les travaux d’Alvarez, Cardaliaguetet Monneau [2] ou de Barles et Ley [20] sur la dynamique des dislocations (voirplus bas) montrent notamment que des estimations de périmètre sont une des clésdu problème. De telles estimations existent dans le cas de vitesses positives (voir[2, 28]): elles sont basées sur des arguments de contrÃle optimal qui échouent dansle cas où la vitesse peut changer de signe. L’objectif du chapitre 1 est d’étudierce cas. Le point de départ de ce travail est la formule de Hadamard, qui est uneformule de dérivation sous le signe intégral sur un domaine évoluant avec le temps.Plus précisément, soit Γ(t) = ∂K(t) pour t ∈ [0, T ] un front régulier et borné pourtout temps. Notons u = 1K la fonction indicatrice de l’ensemble K. La méthodelevel-set montre que, si Vx,t désigne la vitesse normale du front Γ(t) en x, alors on aformellement, pour tout (x, t) ∈ RN × [0, T ],

ut(x, t) = Vx,t |Du(x, t)|.

Alors pour tout φ ∈ C1(RN × [0, T ]; R), on a de façon toujours très formelle

d

dt

∫

K(t)

φ(x, t) dx =d

dt

∫

RN

φ(x, t) 1K(t)(x) dx

=d

dt

∫

RN

φ(x, t) u(x, t) dx

=

∫

RN

(φt(x, t) u(x, t) + φ(x, t) ut(x, t)) dx

=

∫

K(t)

φt(x, t) dx+

∫

RN

φ(x, t)Vx,t |Du(x, t)| dx.

Mais comme on l’a vu plus haut (voir (1.7)),

|Du(·, t)| × LN = HN−1⌊∂K(t),

de sorte que l’on a

d

dt

∫

K(t)

φ(x, t) dx =

∫

K(t)

φt(x, t) dx+

∫

∂K(t)

Vx,t φ(x, t) dHN−1(x). (2.1)

C’est la formule de Hadamard. On y voit apparaître le terme habituel de dérivationsous le signe intégral, mais aussi un terme de bord faisant intervenir la vitessenormale du front. L’objet du chapitre 1 est de montrer une formule de ce type pourles solutions de l’équation eikonale

Vx,t = c(x, t),

dans le cadre des solutions de viscosité, et donc sans régularité a priori sur K(t).L’objectif est en fait d’obtenir de la régularité sur K(t). Pour expliquer cela, faisonsà nouveau un raisonnement formel: appliquons la formule de Hadamard (2.1) à

12

2. Formulations intégrales de l’équation eikonale

φ = sgn(c), la fonction signe de c, en supposant que Dc(x, t) 6= 0 si c(x, t) = 0. Onobtient alors∫

∂K(t)

c(x, t)φ(x, t) dHN−1(x) =d

dt

∫

K(t)

φ(x, t) dx−∫

K(t)

φt(x, t) dx,

c’est-à-dire∫

∂K(t)

|c(x, t)| dHN−1(x) =d

dt

∫

K(t)

sgn(c(x, t)) dx−∫

K(t)

∂sgn(c)

∂t(x, t) dx.

Mais sgn(c) = 1c>0 − 1c<0, et comme on l’a vu précédemment avec u = 1K ,

∂1c>0∂t

(x, t) =ct(x, t)

|Dc(x, t)|︸︷︷︸vitesse de ∂c(·,t)>0

|D1c>0(x, t)|,

tandis que∂1c<0∂t

(x, t) = − ct(x, t)

|Dc(x, t)|︸︷︷︸vitesse de ∂c(·,t)<0

|D1c<0(x, t)|.

De plus |D1c(·,t)>0| × LN = |D1c(·,t)<0| × LN = HN−1⌊c(·,t)=0. On obtient donc

∫

∂K(t)

|c(x, t)| dHN−1(x) =d

dt

∫

K(t)

sgn(c(x, t)) dx−2

∫

K(t)∩c(·,t)=0

ct(x, t)

|Dc(x, t)| dHN−1(x).

De cette égalité, on déduit une estimation du terme∫ T

0

∫

∂K(t)

|c(x, t)| dHN−1(x)dt,

qui est une estimation intégrale en temps liée au périmètre du front ∂K(t): il s’agitdonc bien d’un résultat de régularité des solutions de l’équation eikonale.

Résultats obtenus

En reproduisant ces arguments rigoureusement, nous pouvons:

1. Démontrer des formulations intégrales pour l’équation eikonale Vx,t = c(x, t),inspirées de la formule de Hadamard. On donnera une formulation pour la notionde sous-solution (voir le Théorème 3.1 page 53) ainsi qu’une formulation pour lanotion de sur-solution (voir le Théorème 6.1 page 76).

2. Prouver que ces formulations caractérisent respectivement les notions de sous-solution et sur-solution de cette équation: il s’agit des Théorèmes 4.1 page 60 et 6.3page 76.

3.Montrer que les fronts solutions d’équations eikonales sont de périmètre localementfini dans l’ensemble c 6= 0 pour presque tout temps, quelle que soit la donnéeinitiale et le signe de c.

4. Fournir une borne explicite liée au périmètre des K(t), très ressemblante à celleobtenue formellement. Les résultats précis des points 3 et 4 font l’objet du Théorème7.1 page 77.

13


3 Equations non-locales sans principe de compara-

ison

Dans toute la suite, on s’intéresse à des lois d’évolution pour lesquelles la vitessenormale Vx,t de ∂K(t) au point x est fonction de l’ensemble K(t) lui-même, voiremême de tous les ensembles K(s) pour s ∈ [0, T ]. Ces équations sont donc ditesnon-locales, par opposition au cas où la vitesse Vx,t ne dépend que des données enx, t. Commençons par expliciter la difficulté principale liée à ces équations.

3.1 Non respect du principe de comparaison

Le point clé dans toutes les démonstrations d’existence et d’unicité de solu-tions aux problèmes de propagations de fronts, et plus généralement de solutions deviscosité d’équations paraboliques, est le principe de comparaison évoqué dans lasection 1.2, qui correspond géométriquement à un principe d’inclusion sur les fronts:si u est une sous-solution de viscosité de ut = H(x, t,Du,D2u) dans RN× ]0, T [ etv en est une sur-solution, alors

u∗(x, 0) ≤ v∗(x, 0) dans RN ⇒ u∗(x, t) ≤ v∗(x, t) dans R

N × [0, T [.

C’est par exemple bien le cas de l’équation eikonale, du mouvement par courburemoyenne et de toutes les équations locales classiques. L’idée en est la suivante:supposons pour simplifier que u et v sont continues avec u(·, 0) ≤ v(·, 0), et soit tle premier instant où u(x, t) = v(x, t) pour un certain x ∈ RN . En supposant u etv assez régulières, les conditions d’optimalité du premier et second ordre montrentque

Du(x, t) = Dv(x, t) et D2u(x, t) ≤ D2v(x, t).

L’équation étant parabolique, on a donc

ut(x, t) = H(x, t,Du(x, t), D2u(x, t)) ≤ H(x, t,Dv(x, t), D2v(x, t)) = vt(x, t),

et donc la propriété u(x, t) ≤ v(x, t) est conservée aux temps ultérieurs. On a deplus jusqu’au temps t,

u(·, s) ≥ 0 ⊂ v(·, s) ≥ 0, i.e. 1u(·,s)≥0 ≤ 1v(·,s)≥0.

Demander un principe de comparaison pour les équations non-locales, dont la vitesseest donnée par H [1u≥0], reviendrait donc en appliquant le raisonnement précédentà pouvoir garantir, χ1 et χ2 étant fixés dans L∞(RN × [0, T ]; [0, 1]), que

χ1 ≤ χ2 ⇒ H [χ1] ≤ H [χ2].

Ceci est vrai par exemple pour l’équation non-locale Vx,t = V ol(K(t)) pour laquelle

H [χ](t, p) =

(∫

RN

χ(x, t) dx

)|p|,

mais cela est faux en général. Pour illustrer ce point nous allons présenter deuxmodèles que nous rencontrerons régulièrement par la suite.

14

3. Equations non-locales sans principe de comparaison

3.2 Deux modèles non-locaux

3.2.1 La dynamique des dislocations

Une dislocation est un défaut linéaire qui correspond à une discontinuité dansl’organisation de la structure cristalline des métaux. Ces lignes se déplacent dansdes plans définis et sont la principale explication microscopique des déformationsplastiques observées dans les cristaux à l’échelle macroscopique. Nous renvoyonsprincipalement à [4, 66] et à la thèse de Forcadel [42] pour une présentation plusdétaillée de ce modèle. Dans le modèle de [66], la vitesse normale d’une ligne dedislocation ∂K(t) est donnée par la loi non-locale suivante:

Vx,t = c0(·, t) ⋆ 1K(t)(x) + c1(x, t)

=

∫

K(t)

c0(x− y, t) dy + c1(x, t),(3.1)

d’équation level-set associéeut(x, t) =

[c0(·, t) ⋆ 1u(·,t)≥0(x) + c1(x, t)

]|Du(x, t)| dans RN×]0, T [,

u(x, 0) = u0 dans RN .

(3.2)

Le noyau c0, qui donne le terme non-local par convolution en espace avec 1K(t), estune fonction symétrique qui dépend du matériau; le terme c0(·, t) ⋆ 1K(t) représentela force provenant du champ élastique crée par la ligne de dislocation, et donc laforce exercée par la ligne de dislocation sur elle-même. Le terme c1 représente quantà lui les contraintes extérieures. Le noyau c0 est de moyenne nulle, et en particulierchange de signe, de sorte que le principe de comparaison n’est pas vérifié dans cemodèle: on peut avoir χ1 ≤ χ2 sans que c0(·, t) ⋆χ1(·, t) ≤ c0(·, t) ⋆χ2(·, t) pour toutt ∈ [0, T ].

Un autre modèle possible pour la dynamique des dislocations, qui prend enconsidération la tension de surface de la ligne de dislocation, consiste à ajouter,dans l’expression de la vitesse Vx,t, la courbure moyenne de ∂K(t) en x,

Hx,t = Tr(Ax,t) = −N∑

i=1

∂νi

∂xi

(x, t)

(rappelons que Ax,t désigne la matrice courbure, et ν(x, t) le vecteur normal extérieurunitaire, de K(t) en x). L’équation géométrique qui en résulte est donc

Vx,t = Hx,t + c0(·, t) ⋆ 1K(t)(x) + c1(x, t), (3.3)

et son équation level-set associée estut =

[div(

Du|Du|

)+ c0(·, t) ⋆ 1u(·,t)≥0(x) + c1(x, t)

]|Du| dans RN×]0, T [,

u(·, 0) = u0 dans RN .(3.4)

A nouveau, c’est un problème non-local qui ne respecte pas le principe de compara-ison.

15


3.2.2 Le système de FitzHugh-Nagumo

On s’intéressera aussi de près au système suivant,

ut = α(v)|Du| dans RN×]0, T [,

vt − ∆v = g+(v)1u≥0 + g−(v)(1 − 1u≥0) dans RN×]0, T [,

u(·, 0) = u0, v(·, 0) = v0 dans RN ,

(3.5)

obtenu comme système limite quand ε → 0 du système de FitzHugh-Nagumo ap-paraissant dans les phénomènes de propagation d’influx dans les neurones et encinétique chimique:

uεt − ε∆uε = ε−1f(uε, vε),

vεt − ∆vε = g(uε, vε)

dans RN×]0, T [, où, pour (u, v) ∈ R

2,f(u, v) = u(1 − u)(u− a) − v (0 < a < 1),

g(u, v) = u− γv (γ > 0).

Les fonctions α, g+ et g− apparaissant dans (3.5) sont liées à f et g (voir [48, 72]).

Le système (3.5) est un système couplé dont la première équation décrit le mou-vement d’un front Γ(t) = u(·, t) = 0 qui évolue avec une vitesse α(v), v étantelle-même liée à Γ(t) comme solution d’une équation de réaction-diffusion dont lasource dépend de l’interface Γ(t). Cependant ce système se met sous la forme (1.4):pour χ ∈ L∞(RN × [0, T ]; [0, 1]), soit v la solution de

vt − ∆v = g+(v)χ+ g−(v)(1 − χ) dans RN×]0, T [,

v(·, 0) = v0 dans RN ,(3.6)

et soit c[χ](x, t) = α(v(x, t)). Alors (3.5) est équivalent àut(x, t) = c[1u≥0](x, t)|Du(x, t)| dans RN×]0, T [,

u(x, 0) = u0 dans RN .(3.7)

On remarque que dans ce modèle la solution v de (3.6), et donc la vitesse c[χ], dépendde χ de manière non-locale en espace mais aussi en temps: en effet la résolutionexplicite de l’équation de la chaleur (3.6) montre que pour tout (x, t) ∈ RN×]0, T [,

v(x, t) =

∫

RN

G(x− y, t) v0(y) dy

+

∫ t

0

∫

RN

G(x− y, t− s) [g+(v)χ+ g−(v)(1 − χ)](y, s) dyds,

(3.8)

où G est la fonction de Green définie par

G(y, s) =1

(4πs)N/2e−

|y|2

4s . (3.9)

De plus, la fonction α n’ayant pas de propriété de croissance, ce système ne vérifiepas le principe de comparaison.

16

3. Equations non-locales sans principe de comparaison

3.3 Solutions faibles d’équations non-locales

Du fait de l’absence de principe de comparaison, les techniques classiques deconstruction d’une solution de viscosité à (1.4) échouent. Une possibilité pour con-tourner cette difficulté consiste à adapter la notion de solution, en proposant unenotion de solution faible de (1.4). C’est cette notion, centrale dans cette thèse,que nous présentons maintenant. L’idée de solution faible, déjà utilisée dans [48] et[72], et plus récemment dans [16], apparaît naturellement en étudiant la stabilité desolutions de (3.2) par exemple: considérons une suite (uε) de solutions de

(uε)t(x, t) =[c0(·, t) ⋆ 1uε(·,t)≥0(x) + c1(x, t)

]|Duε(x, t)| dans RN×]0, T [,

uε(x, 0) = u0(x) dans RN .

Sous de bonnes hypothèses sur c0 et c1, on obtient des bornes uniformes sur (uε)t et|Duε|, et l’on peut supposer par le théorème d’Ascoli que uε → u dans C0(RN×[0, T ])lorsque ε → 0 (à sous-suite près). En particulier

1uε(x,t)≥0 −→ε→0

1 si u(x, t) > 0,

0 si u(x, t) < 0.

En revanche, on ne peut rien conclure quand u(x, t) = 0.

Cela dit pour tout ε, 1uε≥0 est à valeurs dans [0, 1]. Par le théorème de Banach-Alaoglu, on peut supposer que 1uε≥0 χ dans L∞(RN × [0, T ]; [0, 1]) pour latopologie faible-∗ (à sous-suite près). On obtient alors formellement à la limite

ut(x, t) = [c0(·, t) ⋆ χ(·, t)(x) + c1(x, t)] |Du(x, t)| dans RN×]0, T [,

u(x, 0) = u0(x) dans RN ,(3.10)

et1u>0 ≤ lim inf 1uε≥0 ≤ χ ≤ lim sup 1uε≥0 ≤ 1u≥0. (3.11)

Plusieurs difficultés se posent tout de même ici:

1. Dans l’équation limite (3.10), la vitesse c0(·, t)⋆χ(·, t)(x), si elle est assez régulièreen espace (en effet elle est obtenue par convolution avec un noyau assez régulier),est seulement mesurable en temps. Pour ce type d’équations, la notion de solutionde viscosité doit être adaptée pour prendre en compte cette absence de régularité entemps: il s’agit de la notion de solution de viscosité L1 introduite dans la Définition1.6. Nous renvoyons le lecteur à [50, 61, 62, 27, 26] pour une présentation complètede la théorie des solutions de viscosité L1.

2. Le passage à la limite formel doit être justifié pour des Hamiltoniens H [χn] lorsquela convergence de (χn) a seulement lieu dans L∞(RN × [0, T ]; [0, 1]) pour la topologiefaible-∗. Une réponse à ce problème est donnée par un résultat de stabilité de Barles[15] prenant en compte ce type de convergence sous l’hypothèse minimale suivante:

(H) :χn n→∞

χ dansL∞ faible-∗ ⇒∫ t

0

H [χn](x, s, p, A)ds −→n→∞

∫ t

0

H [χ](x, s, p, A)ds

17


localement uniformément par rapport à x, t, p, A.

3. Si u désigne la solution de (3.10), alors u > 0 peut être très différent deu ≥ 0: en effet u = 0 peut être de mesure non nulle, ou d’intérieur non vide. Ils’agit du phénomène de “fattening”, qui empêche d’identifier χ comme 1u≥0 dansl’inégalité (3.11), ce qui est pourtant l’objectif voulu pour fournir une solution de(3.2) ou des autres équations non-locales du type (1.4). Le phénomène de fatteningest un problème typique et inévitable de l’approche par lignes de niveau pour lespropagations de fronts, y compris celles gouvernées par des lois locales. Par exempleil a lieu instantanément lors de l’évolution par courbure moyenne d’un front initialen forme de ∞. Dans le raisonnement précédent, ce phénomène souligne clairementla difficulté à obtenir ne serait-ce que l’existence d’une solution de viscosité de (1.4).

Malgré tout, l’argument heuristique précédent permet de définir une notion desolution faible de (1.4):

Définition 3.1 (Solution faible).Soit u : R

N × [0, T ] → R une fonction continue. On dit que u est solution faiblede (1.4) s’il existe χ ∈ L∞(RN × [0, T ]; [0, 1]) tel que:

1. u est solution de viscosité L1 deut(x, t) = H [χ](x, t,Du,D2u) dans RN×]0, T [,

u(x, 0) = u0(x) dans RN .(3.12)

2. Pour presque tout t ∈ [0, T ],

1u(·,t)>0 ≤ χ(·, t) ≤ 1u(·,t)≥0 presque partout dans RN .

De plus on dit que u est solution de viscosité classique de (1.4) si on a, pour presquetout t ∈ [0, T ],

1u(·,t)>0 = 1u(·,t)≥0 presque partout dans RN .

4 Existence de solutions faibles pour des équations

non-locales générales

Dans le chapitre 2, on s’intéresse pour commencer à l’existence de telles solutionsfaibles aux équations non-locales (1.4). La méthode qui suit n’est pas nouvelle: elleapparaît déjà dans [48] pour le système de FitzHugh-Nagumo. Nous l’avons utiliséepour obtenir des résultats d’existence de solutions faibles pour des équations non-locales générales. Considérons l’application multi-valuée

ξ : χ ∈ L∞(RN × [0, T ]; [0, 1])

7→ χ′; 1u(·,t)>0 ≤ χ′(·, t) ≤ 1u(·,t)≥0 pour presque tout t ∈ [0, T ],

où u est la solution de viscosité L1 de ut = H [χ](x, t,Du,D2u)

avec condition initiale u(·, 0) = u0.

18

4. Existence de solutions faibles pour des équations non-locales générales

Si ξ a un point fixe χ, dans le sens que χ ∈ ξ(χ), alors la fonction u correspondanteest une solution faible de (1.4). L’existence de solutions faibles de (1.4) résultedonc par exemple du théorème de point fixe de Kakutani sur les applications multi-valuées:

Théorème 4.1 (Point fixe de Kakutani [10]).Soit K ⊂ X un compact convexe non-vide d’un espace localement convexe séparé,

et soit ξ : K → P(K) une application à valeurs non-vides fermées convexes. Onsuppose que ξ est semi-continue supérieurement, c’est-à-dire que

χn → χ

χ′n ∈ ξ(χn) → χ′ ⇒ χ′ ∈ ξ(χ).

Alors ξ a un point fixe dans K: il existe χ ∈ K tel que χ ∈ ξ(χ).

Dans notre cas, nous pourrons utiliser essentiellement K = L∞(RN×[0, T ]; [0, 1])dans X = L∞(RN × [0, T ]; R) muni de la topologie faible-∗. L’hypothèse de semi-continuité supérieure provient précisément du résultat de stabilité de Barles cité plushaut et de l’utilisation de (H). L’application de ce théorème de point fixe fournitdonc une solution faible de (1.4). Plus précisément, les principaux résultats duchapitre 2 sont les suivants:

Résultats obtenus

1. Si pour toute fonction χ fixée, le Hamiltonien H [χ] vérifie certaines hypothèsesde régularité – garantissant notamment un principe de comparaison pour l’équationlocale ut = H [χ](x, t,Du,D2u) – et si l’application χ 7→ H [χ] vérifie l’hypothèsede continuité (H), alors il existe au moins une solution faible de (1.4). Nous don-nons en fait deux résultats d’existence: le premier correspond au cas favorable oùH [χ] est défini quelle que soit la fonction χ, et notamment quelle que soit la taillede son support; c’est le Théorème 4.1 page 85. Le second résultat (Théorème 5.1page 92) s’applique aux équations pour lesquelles il est nécessaire que le supportde χ soit borné pour que H [χ] soit défini: c’est le cas par exemple des équationsfaisant intervenir le volume de l’ensemble K(t). Dans les deux cas, il s’agit d’unrésultat pour des équations très générales: ces équations ne sont pas nécessairementgéométriques (bien que dans les applications, on ait à l’esprit des modèles de prop-agations de fronts, donc des équations géométriques), et le Hamiltonien H [χ] peutéventuellement dépendre des valeurs de χ(·, s) pour tout temps s ∈ [0, T ], passé oufutur.

2. En application du premier résultat d’existence, on montre l’existence de solutionsfaibles pour la dynamique des dislocations (3.2) et le problème de FitzHugh-Nagumo(3.7). Il s’agit de résultats déjà connus (cf. [16] et [48, 72]) que nous retrouvonsfacilement grâce au théorème général. En application du second résultat, on traiteles équations dépendant du volume de K(t).

19


5 Dynamique des dislocations avec terme de cour-

bure moyenne

On obtiendrait aussi facilement l’existence de solutions faibles pour la dynamiquedes dislocations avec un terme de courbure moyenne (3.4) en application du théorèmegénéral d’existence (Théorème 4.1 page 85). Cela dit, l’intérêt du terme de courburemoyenne dans le modèle de dynamique des dislocations, outre le fait qu’il prend encompte la tension de surface de la ligne de dislocation, est son aspect régularisant,de sorte que l’on s’attend à obtenir mieux qu’une solution faible telle que décritedans la Définition 3.1. C’est l’objet du chapitre 3 que nous présentons maintenant.

Dans ce chapitre, on construit une solution faible de (3.4) en cherchant χ(·, t) nonpas sous la forme idéale 1u(·,t)≥0, mais tout de même sous la forme d’une indicatrice1E(t), l’application t 7→ 1E(t) ayant une certaine régularité.

On va construire E comme mouvement minimisant pour la loi d’évolution(3.3): ceci consiste à trouver E comme le flot qui minimise l’énergie associée àcette loi. Cette idée est due à Almgren, Taylor et Wang [1], et indépendamment àLuckhaus et Sturzenhecker [55], pour le mouvement par courbure moyenne. Pourexpliquer cette approche, prenons le problème à l’envers: on va discrétiser l’équation

Vx,t = Hx,t + c0(·, t) ⋆ 1K(t)(x) + c1(x, t) (5.1)

en temps. Soit donc h un pas de temps. On cherche une suite d’ensembles Eh(k),pour tout k ∈ N tel que kh ≤ T , dont l’évolution avec k soit une discrétisation de(5.1). Supposons cette suite construite. Pour x ∈ ∂Eh(k + 1) tel que x /∈ Eh(k),

d∂Eh(k)(x)

h

est une approximation de la vitesse normale de ∂Eh(k+1) en x. De la même façon,si x ∈ Eh(k),

−d∂Eh(k)(x)

h

est une approximation de la vitesse normale de ∂Eh(k + 1) en x. On cherche doncà construire une suite d’ensembles Eh(k) tels que pour tout x ∈ ∂Eh(k + 1),

±d∂Eh(k)(x)

h= Hx,(k+1)h + c0(·, (k + 1)h) ⋆ 1Eh(k+1)(x) + c1(x, (k + 1)h), (5.2)

où l’on choisit le signe + si x /∈ Eh(k), le signe − sinon; Hx,(k+1)h désigne la courburemoyenne de Eh(k+1) en x. Ceci correspond à une discrétisation implicite en tempsde (5.1).

Connaissant Eh(k), on construit alors Eh(k+1) en voyant (5.2) comme équation

20

5. Dynamique des dislocations avec terme de courbure moyenne

d’Euler-Lagrange formelle associée à la minimisation de la fonctionnelle

E 7→ F(h, k + 1, E, Eh(k))

= P (E) +1

h

∫

E∆Eh(k)

d∂Eh(k)(x) dx

−∫

E

(1

2c0(·, (k + 1)h) ⋆ 1E(x) + c1(x, (k + 1)h)

)dx,

qui contient l’énergie du système ainsi qu’un terme de pénalisation correspondant àla vitesse discrète. Dans ce terme, pour E,F ⊂ RN , E∆F = (E ∪ F ) \ (E ∩ F ) estla différence symétrique de E et F .

On s’attend à ce que tout minimiseur de cette fonctionnelle soit solution de (5.2).L’idée sera ensuite de passer Eh à la limite lorsque h → 0 en choisissant, pour toutt ∈ [0, T ] fixé, k = [t/h], de sorte que kh → t quand h → 0. On obtient alors unmouvement minimisant t 7→ E(t) associé à la fonctionnelle F .

La fonctionnelle F est définie sur l’ensemble P des sous-ensembles bornés de RN

de périmètre fini. Sur cet ensemble, on utilise la distance

δ(E,F ) = LN(E∆F ) = ‖1E − 1F‖L1(RN ).

On dit alors que En −→n→+∞

E dans L1(RN) lorsque δ(En, E) −→n→+∞

0.

Définition 5.1 (Mouvement minimisant).Soit E0 ∈ P. On dit que E : [0, T ] → P est un mouvement minimisant associé

à F de condition initiale E0 si il existe hn → 0+ et des ensembles Ehn(k) ∈ P pourtout k ∈ N verifiant khn ≤ T , tels que:

1. Ehn(0) = E0.

2. Pour tous k, n ∈ N tels que (k + 1)hn ≤ T ,

Ehn(k + 1) minimise la fonctionnelle E → F(hn, k + 1, E, Ehn(k)).

3. Pour tout t ∈ [0, T ], Ehn([t/hn]) → E(t) dans L1(RN) quand n→ +∞.

Si E est un minimiseur de E → F(h, k+ 1, E, Eh(k)), on montre en considérantde petites perturbations locales de E que l’équation discrète (5.2) est vérifiée en toutpoint régulier de ∂E. La question de la régularité des minimiseurs de la fonctionnelleF est donc fondamentale. Elle résulte de la théorie des surfaces minimales: dansla fonctionnelle F , le terme dominant est le périmètre (il est d’ordre 2, car lié à lacourbure moyenne). Tout se passe donc à peu de choses près comme si on considéraitseulement la fonctionnelle périmètre. Or il se trouve que les (quasi-)minimiseurs dupérimètre sont réguliers partout en dimension inférieure à 7, et “presque partout”sinon. Dans notre cas, on obtient la même régularité, grâce à la théorie des courants.

Les résultats obtenus, qui font l’objet du chapitre 3, sont les suivants:

Résultats obtenus

21


Sous des hypothèses adaptées sur c0 et c1, on a:

1. Si LN(E0) = 0, alors il existe des mouvements minimisants HËlderiens asso-ciés à F de condition initiale E0, c’est-à-dire qu’il existe au moins un mouvementminimisant E et une constante C > 0 tels que pour tous t, s ∈ [0, T ],

δ(E(t), E(s)) ≤ C|t− s| 1N+1 .

Ce résultat correspond au Théorème 1.3 page 100.

2. La solution correspondante deut =

[div(

Du|Du|

)+ c0(·, t) ⋆ 1E(t)(x) + c1(x, t)

]|Du| dans RN×]0, T [,

u(·, 0) = u0 dans RN ,

est une solution faible de (3.4) avec χ = 1E : il s’agit du Théorème 1.4 page 101. Cerésultat utilise très fortement l’équation d’Euler-Lagrange (5.2), elle-même obtenuegrâce à la théorie de la régularité [49]. Il est inspiré des travaux de Chambolle[33] sur la consistance entre les notions de mouvement minimisant et de solution deviscosité pour le mouvement par courbure moyenne.

3. Si E0 est un compact dont le bord est de classe C3+α, alors il existe t0 > 0 et uneunique évolution régulière Er(t)0≤t≤t0 telle que ∂Er(t) est de classe C3+α pourtout t avec Er(0) = E0, et dont la vitesse normale est donnée par (3.3): voir leThéorème 1.6 page 101.

De plus, tout mouvement minimisant E associé à F de condition initiale E0

vérifie E(t) = Er(t) pour tout t ∈ [0, t0] et presque partout dans RN : c’est l’objet

du Théorème 1.5 page 101.

6 Unicité de solutions faibles pour les deux prob-

lèmes modèle

Les résultats des chapitres 2 et 3 concernent comme on vient de le voir l’existencede solutions faibles d’équations non-locales de la forme (1.4). Nous nous sommesaussi naturellement intéressés à leur unicité dans le cas des deux problèmes modèleprésentés plus haut, la dynamique des dislocations (3.2) et le système de FitzHugh-Nagumo (3.5) équivalent à (3.7). C’est le problème abordé dans le chapitre 4, quenous présentons maintenant.

Les deux équations modèle (3.2) et (3.7) sont des cas particuliers de (1.4) dupremier ordre, de la forme d’une équation eikonale non-locale

ut(x, t) = c[1u≥0](x, t) |Du(x, t)| dans RN×]0, T [,

u(·, 0) = u0 dans RN .(6.1)

La question de l’unicité est difficile: elle est liée au phénomène de fattening, qui peutavoir lieu pour les équations du premier ordre dès que la vitesse change de signe.

22

6. Unicité de solutions faibles pour les deux problèmes modèle

Barles, Cardaliaguet, Ley et Monneau [16] ont ainsi construit un contre-exempleexplicite dans le cas de l’équation des dislocations. Pour l’équation (6.1) générale,lorsqu’au contraire il existe δ > 0 tel que pour tout χ ∈ L∞(RN × [0, T ]; [0, 1]),on a c[χ] ≥ δ, il est alors connu (voir Ley [53]) que toute solution faible est enfait une solution classique, car l’ensemble de niveau u = 0 est de mesure nulle:le phénomène de fattening n’a pas lieu. La question de l’existence de solutionsclassiques a donc une réponse positive. Cependant, cette hypothèse de positivité dela vitesse ne remplace pas un principe de comparaison, et la question de l’unicitéreste entière.

Malgré tout, des techniques spécifiques au cas de vitesses positives ont permisà Alvarez, Cardaliaguet et Monneau [2] et Barles, Ley [20] de prouver existenceet unicité pour l’équation des dislocations (par des techniques différentes). Leursrésultats sont synthétisés et améliorés dans [16]. Le point commun des argumentsde [2, 20, 16] est de demander toutefois une régularité assez forte en espace pourles vitesses c[χ] (typiquement C1,1). En ce qui concerne le système de FitzHugh-Nagumo (3.5), l’unicité était jusqu’à présent une question ouverte, y compris dansle cas d’une vitesse positive. De plus pour ce système, les vitesses c[χ] n’ont pas larégularité C1,1 et les techniques de [2, 20, 16] ne peuvent être appliquées en l’état.Le but du chapitre 4 est de prouver l’unicité d’une solution classique pour ces deuxéquations, dans le cas de vitesses positives. Pour la dynamique des dislocations,l’amélioration par rapport aux résultats précédents est la moindre régularité exigéesur les vitesses.

Dans les deux cas, c[χ] est obtenu par une procédure de convolution en espace(dans le cas de (3.2)) ou en espace-temps (dans le cas de (3.7), comme le montre laformule de représentation (3.8)). On voit donc facilement que, bien que χ ne soitrégulière ni en temps ni en espace, la vitesse c[χ] est toujours au moins Lipschitzienneen x; en revanche, elle n’a pas nécessairement de régularité en t. Plus précisément,la vitesse c[χ] = c satisfait les hypothèses suivantes:

(H1) Il existe C, c, c > 0 tels que pour tout χ ∈ L∞(RN × [0, T ]; [0, 1]), pour toutx ∈ RN , t 7→ c(x, t) est mesurable et pour tous x, y ∈ RN et t ∈ [0, T ],

|c(x, t) − c(y, t)| ≤ C|x− y|,0 < c ≤ c(x, t) ≤ c.

Par exemple pour l’équation des dislocations (3.2), l’hypothèse de positivité estvérifiée si c1(x, t) ≥ ‖c0(·, t)‖1 + c pour tout (x, t) ∈ RN × [0,+∞[, ce qui garantitque

∀K ⊂ RN , c0(·, t) ⋆ 1K(x) + c1(x, t) ≥ −‖c0(·, t)‖1 + c1(x, t) ≥ c.

On fera de plus une hypothèse de régularité sur l’ensemble initial K0 = u0 ≥ 0:(H2) La fonction u0 est Lipschitzienne et il existe η0 > 0 tel que

−|u0(x)| − |Du0(x)| + η0 ≤ 0 dans RN au sens viscosité.

Ces hypothèses assurent comme mentionné plus haut que le phénomène de fatteningn’a pas lieu: la solution u de (3.12) a la propriété que l’ensemble u = 0 est de

23


mesure nulle (voir Ley [53]). De plus, (H1) et (H2) impliquent l’existence d’uneborne inférieure de gradient |Du| ≥ η valable sur l’ensemble x ∈ RN ; |u(x, t)| ≤ ηpour tout t ∈ [0, T ] et η assez petit. (cf. [53]).

Donnons maintenant une esquisse de la preuve d’unicité pour cerner les difficultésdu problème: soient donc u1 et u2 deux solutions classiques de (6.1). Pour 0 < τ ≤ T,on définit

δτ := supRN×[0,τ ]

|u1(x, t) − u2(x, t)|.

Comme u1(·, 0) = u2(·, 0), on a δτ ≤ η pour τ assez petit. Une estimation classiquemontre de plus que u1−u2 est contrÃlé en fonction de la différence des deux vitessesc[1u1≥0] − c[1u2≥0]: plus précisément

δτ ≤ ‖Du0‖∞eCτ

∫ τ

0

‖(c[1u1≥0] − c[1u2≥0])(·, t)‖∞ dt.

Dans le cas de la dynamique des dislocations (3.2), en supposant de plus que c0 estborné avec ‖c0‖∞ ≤ c, on poursuit cette estimation par


∫ τ

0

‖c0(·, t) ⋆ (1u1(·,t)≥0 − 1u2(·,t)≥0)‖∞ dt

≤ c ‖Du0‖∞eCT

∫ τ

0

∫

RN

|1u1≥0 − 1u2≥0|(x, t) dxdt.(6.2)

Dans le cas du système de FitzHugh-Nagumo (3.5), en notant v1, v2 les solutions de(3.6) correspondant à χ = 1u1≥0, 1u2≥0 respectivement, on obtient


∫ τ

0

‖(α(v1) − α(v2))(·, t)‖∞ dt

≤ ‖Du0‖∞eCT ‖α′‖∞∫ τ

0

‖(v1 − v2)(·, t)‖∞ dt.

Mais on montre en utilisant la formule de représentation (3.8) qu’il existe une con-stante L > 0 telle que pour tout (x, t) ∈ RN × [0, T ],

|v1(x, t) − v2(x, t)|

≤ L

∫ t

0

∫

RN

G(x− y, t− s) |1u1≥0 − 1u2≥0|(y, s) dyds.(6.3)

De plus dans les deux cas, on a par définition de δτ ,

|1u1≥0 − 1u2≥0| ≤ 1−δτ≤u1≤0 + 1−δτ≤u2≤0 dans RN × [0, τ ].

On se rend donc compte que, pour poursuivre ces majorations, une étape clé estd’obtenir des estimations sur le volume d’ensembles du type a ≤ ui(·, t) ≤ b avec−η ≤ a < b ≤ η, en fonction de b− a: elles permettraient de prouver que

δτ ≤ L(τ) δτ ,

24

6. Unicité de solutions faibles pour les deux problèmes modèle

pour une certaine constante L(τ) qui tend vers 0 lorsque τ → 0, et donc de conclure,pour τ assez petit, que δτ = 0, c’est-à-dire l’unicité en temps court. Reproduisantl’argument, on obtient l’unicité sur [0, T ]. Remarquons de plus que de telles estima-tions sont liées au contrÃle du périmètre des lignes de niveau u(·, t) = α pour αproche de zéro: en effet, formellement, en utilisant la formule de la coaire (voir [38])et la borne inférieure de gradient |Du| ≥ η, on obtient

∫

RN

1a≤u(·,t)≤b(x) dx =

∫ b

a

∫

u(·,t)=s|Du(x, t)|−1 dHn−1(x)ds

≤ b− a

ηsup

a≤s≤bP (u(·, t) = s). (6.4)

Dans les articles [2] et [20] sur la dynamique des dislocations, ces estimations depérimètre sont obtenues par des bornes sur la courbure de u(·, t) = α correspon-dant à une condition de boule intérieure des ensembles u(·, t) ≥ α, ce qui demandeune régularité assez forte sur la dépendance en x de c[χ]. Cette régularité n’est pasdisponible en général: par exemple elle fait défaut pour le système de FitzHugh-Nagumo.

Finalement, pour la dynamique des dislocations, lorsque le noyau c0 est borné,(6.2) montre que des estimations dans L1([0, T ]) sur le volume des ensembles dutype a ≤ u(·, t) ≤ b suffisent. Elles sont de plus relativement faciles à obtenirdirectement.

En revanche, pour le système de FitzHugh-Nagumo, le noyau de Green définipar (3.9) n’est pas borné mais intégrable, et des estimations dans L1([0, T ]) nesuffisent pas; on a besoin d’estimations dans L∞([0, T ]) du périmètre des ensemblesu(·, t) = α, comme le suggèrent (6.3) et (6.4). Pour les obtenir, on montre queces ensembles satisfont une condition non pas de boule intérieure, mais de cÃneintérieur, dont on déduit une estimation précise de leur périmètre.

Résultats obtenus

Sous les hypothèses (H1) et (H2), nous avons obtenu les résultats suivants:

1. Les équations (3.2) et (3.7) possèdent une unique solution faible, qui est unesolution classique: voir les Théorèmes 3.1 page 138 et 4.1 page 141. Les preuves deces résultats utilisent les résultats auxiliaires suivants d’intérêt indépendant:

2. Une estimation dans L1([0, T ]) sur le volume des ensembles a ≤ u(·, t) ≤ b dela solution de l’équation eikonale: voir la Proposition 5.5 page 148.

3. Les solutions de l’équation eikonale développent un cÃne intérieur pour touttemps t ∈ [0, T ]. Ce résultat est l’objet du Théorème 5.9 page 152.

4. Un ensemble compact K ayant la propriété de cÃne intérieur vérifie

HN−1(∂K) < +∞

avec une estimation précise de cette mesure: il s’agit du Théorème 5.8 page 149.

25


7 Convergence de schémas d’approximation pour

des équations non-locales

Dans le chapitre 5, nous nous intéressons finalement à l’approximation des équa-tions non-locales du type (1.4). Cette question est source de nombreux travaux:par exemple, Cardaliaguet et Pasquignon [32] et Slepčev [70] ont étudié le casd’équations vérifiant le principe d’inclusion. Alvarez, Carlini, Monneau et Rouy[3] ont proposé un schéma pour la dynamique des dislocations, qui ne vérifie pas leprincipe d’inclusion. Les méthodes de type Fast Marching connaissent également unfort intérêt. On pourra à ce sujet trouver dans les livres de Sethian [68] et de Osheret Fedkiw [63], de nombreux schémas pour une large gamme d’applications. Notretravail est motivé par les résultats d’existence obtenus au chapitre 2, ainsi que par lethéorème de convergence de schémas numériques de Barles et Souganidis [23] dansle cas d’équations locales. On cherche pour notre part à démontrer la convergencede schémas généraux vers une – ou dans les cas favorables vers la – solution faiblede l’équation non-locale (1.4).

Soit h = T/n un pas de temps, pour un certain n ∈ N∗, et soient ∆1, . . . ,∆N

des pas d’espace déterminés par ce choix de h. Pour (i1, . . . , iN) ∈ ZN , on définit lenoeud xi1,...,iN = (i1∆1, . . . , iN∆N), et le pavé en espace

Qi1,...,iN =

N∏

k=1

[(ik − 1/2)∆k, (ik + 1/2)∆k[.

On définit aussi le réseau en espace

Πh =⋃

(i1,...,iN )∈ZN

xi1,...,iN,

et pour x = (x1, . . . , xN) ∈ RN , sa projection sur ce réseau,

xh := ([x1/∆1 + 1/2]∆1, . . . , [xN/∆N + 1/2]∆N) ∈ Πh,

de sorte que si x ∈ Qi1,...,iN , alors xh = xi1,...,iN .

Pour tout x ∈ Πh, kh ≤ T , u : Πh → R et tout χ : Πh × [0, T ] → [0, 1] à supportborné, on définit un Hamiltonien discret Hh[χ](x, kh, u), qui est une approximationde H [χ](x, t,Du,D2u), dépendant des valeurs

χ(xi1,...,iN , lh) pour (i1, . . . , iN) ∈ ZN et 0 ≤ l ≤ k,

u(xi1,...,iN ) pour (i1, . . . , iN) ∈ ZN .

En particulier, on s’intéresse donc aussi aux équations dépendant du passé, commele problème de FitzHugh-Nagumo (3.7).

Les schémas d’approximation que nous considérons sont des schémas explicitesde la forme suivante: pour tout k ∈ N vérifiant (k + 1)h ≤ T , et tout x ∈ Πh, onpose

uh(x, (k + 1)h) = uh(x, kh) + hHh[1uh≥0](x, kh, uh(·, kh)),uh(x, 0) = u0(x).

(7.1)

26

7. Schémas d’approximation pour des équations non-locales

On étend ensuite uh en une fonction constante par morceaux sur RN × [0, T ] en

posant pour chaque (x, t),

uh(x, t) = uh(xh, [t/h]h).

Dans l’esprit du résultat de Barles et Souganidis, on considère ici des schémasconditionnellement monotones, stables et consistants. La monotonie condi-tionnelle tout d’abord signifie que pour toute fonction χ : Πh × [0, T ] → [0, 1] fixéeet k ∈ N tel que kh ≤ T ,

u ≤ v ⇒ u(x)+hHh[χ](x, kh, u) ≤ v(x)+hHh[χ](x, kh, v) pour tout x ∈ Πh.

La stabilité du schéma correspond à l’existence de bornes L∞ uniformes sur lessolutions de (7.1). Insistons sur le fait que ces deux propriétés portent sur le schémaà χ fixé, c’est-à-dire un schéma correspondant à l’équation locale (3.12): on nedemande notamment pas la monotonie du schéma non-local (7.1).

Notre résultat de convergence est basé sur l’idée de la preuve du théorèmed’existence 4.1 du chapitre 2, exposée dans la section 3.3: en effet, la suite (1uh≥0)étant bornée dans L∞(RN × [0, T ]), il en existe une sous-suite, encore indéxée parh par simplicité, convergeant vers un certain χ ∈ L∞(RN × [0, T ]; [0, 1]) pour latopologie de la convergence faible-∗. Il s’agit alors de montrer que (uh) convergelocalement uniformément vers une solution faible de (1.4). Bien entendu, cette con-vergence dépend du bon choix du Hamiltonien discret Hh, c’est-à-dire sa consistancepar rapport à H . Pour avoir l’intuition de la condition de consistance à imposer,il est naturel de se baser sur les résultats de stabilité disponibles pour l’équation(3.12), puisque ceux-ci donnent des conditions sur la convergence des Hamiltoniensqui garantissent la convergence des solutions correspondantes. Or, nous avons déjàsouligné que dans notre cadre, le meilleur résultat de stabilité connu est celui deBarles cité plus haut (voir [15]). La preuve de la convergence du schéma est donctrès naturellement inspirée de la preuve de ce résultat de stabilité. L’hypothèseprincipale de ce théorème (voir page 17) devient, dans notre cadre discret:

χh h→0

χ dans L∞ faible-∗ ⇒ h

[t/h]−1∑

l=0

Hh[χh](xh, s, p, A)ds −→h→0

∫ t

0


localement uniformément en x, t, p, A; c’est notre hypothèse de consistance.

Résultats obtenus

1. Sous cette hypothèse de consistance, et celles de monotonie conditionnelle et destabilité, on montre la convergence de schémas généraux du type (7.1) vers unesolution faible de (1.4): voir le Théorème 2.2 page 165.

2. Nous donnons aussi dans le cas déjà largement étudié de la dynamique des dislo-cations (3.2) et du système de FitzHugh-Nagumo (3.5), des exemples explicites deschémas dont nous prouvons la convergence à l’aide de notre résultat: il s’agit desThéorèmes 3.1 page 171 et 3.2 page 176. Le schéma considéré pour la dynamiquedes dislocations est implémenté numériquement dans [3].

27


8 Conclusion et Perspectives

Les travaux que nous avons présentés concernent tous de près ou de loin l’étuded’équations de propagations de fronts non-locales: nous nous sommes d’abord in-téressés à la régularité des solutions de l’équation eikonale locale, qui est fonda-mentale car elle apparaît naturellement en “gelant” la vitesse non-locale dans lesarguments de type point fixe. A cause de l’absence de principe d’inclusion pour ceséquations non-locales, nous avons ensuite porté notre attention sur une notion desolution faible pour laquelle nous avons étudié existence, unicité et approximationnumérique. L’étude de l’équation eikonale locale joue là aussi un rÃle crucial, etplusieurs résultats d’intérêt indépendant ont été établis pour cette équation: des es-timations sur le volume des ensembles de niveau de la solution, ainsi qu’un résultatsur la création d’un cÃne intérieur à l’ensemble solution, qui fournit de nouvellesestimations de périmètre.

La notion de solution de viscosité et ses techniques spécifiques baignent lestravaux, dans lesquels on recourt toutefois très souvent à des outils plus géométriques:l’idée de cÃne intérieur en est un exemple, ainsi que l’approche par mouvementsminimisants (dans le cas où apparaît la courbure), qui utilise fortement la théoriedes surfaces minimales. Des arguments d’optimisation, de contrÃle optimal et dethéorie géométrique de la mesure sont aussi précieux dans cette étude.

Pour compléter ces travaux, plusieurs pistes sont envisageables:

– La notion de solution faible demande sans doute à être mieux comprise, ouaméliorée: peut-être trop faible, on y perd l’unicité de solution. Notamment, descritères supplémentaires sur la fonction χ apparaissant dans la définition de solutionfaible pourraient être proposés, afin de garantir que cette fonction soit plus prochede la fonction caractéristique 1u≥0. Un premier pas dans cette direction est pro-posé dans le chapitre 3 avec les mouvements minimisants, mais il utilise fortementl’aspect variationnel du problème.

– De ce point de vue, les questions suivantes pourraient être soulevées: l’apparitiondu phénomène de fattening implique-t-il l’absence d’unicité de solution faible ? In-versement, peut-il y avoir défaut d’unicité sans fattening ?

– Pour la dynamique des dislocations avec courbure, il serait intéressant de savoirsi le mouvement minimisant E obtenu est égal à l’ensemble u ≥ 0 y compris encas de fattening, ou si des contre-exemples montrent que même pour cette équationtrès régularisante, l’existence de solution classique est compromise.

– La question de l’unicité pour la dynamique des dislocations avec courburemoyenne est aussi très largement ouverte. Seuls des résultats en temps court sontconnus (voir [43]).

– Du point de vue numérique, la méthode constructive des mouvements min-

28

8. Conclusion et Perspectives

imisants pourrait donner lieu à des simulations pour la dynamique des dislocationsavec courbure moyenne. La méthode des graph cuts, permettant de calculer effi-cacement des minimiseurs de fonctionnelles telles que celle rencontrée au chapitre 3(voir par exemple les travaux de Chambolle [34]), pourrait être exploitée.

29

Notations récurrentes

– Les notations ut(x, t), Du(x, t), D2u(x, t) désignent respectivement la dérivéepartielle par rapport au temps, le gradient et la matrice Hessienne par rapportà l’espace d’une fonction u : RN × [0, T ] → R au point (x, t).

– La fonction indicatrice d’un ensemble A est notée 1A.– La fonction distance à un ensemble fermé non vide F ⊂ RN est notée dF .– Le symbole SN représente l’ensemble des matrices carrées symétriques réelles

de taille N .– La norme et le produit scalaire euclidiens sur RN sont notés | · | et 〈·, ·〉 respec-

tivement.– On note B(x, r) (respectivement B(x, r)) la boule ouverte (respectivement

fermée) de RN de centre x et de rayon r ≥ 0. On trouvera aussi, dans lechapitre 3, les notations Br(x) et Br(x).

– On note Lk la mesure de Lebesgue sur Rk, et ωk la mesure Lk de la bouleunité de Rk.

– La mesure de Hausdorff de dimension N − 1 sur RN est notée HN−1.

– Lorsque f est une fonction mesurable de A ⊂ Rn (n ∈ N∗) à valeurs réelleset 0 < p ≤ +∞, la norme Lp de f est notée ‖f‖p ou ‖f‖Lp(A). On utilis-era aussi régulièrement la notation ‖f‖X pour désigner les normes naturellessur d’autres espaces fonctionnels X. Dans ce cas la définition de ‖f‖X seratoujours précisée.

– Les ensembles de fonctions de classe Ck, Lipschitziennes, intégrables ou es-sentiellement bornées de U ⊂ Rn à valeurs dans V , sous-ensemble de Rm

(n,m ∈ N∗) ou espace fonctionnel, sont notés Ck(U ;V ), Lip(U ;V ), L1(U ;V )

et L∞(U ;V ) respectivement. Si V n’est pas précisé, on sous-entend que V = R.On écrira Ck

c (U ;V ) pour imposer que les fonctions concernées sont à supportcompact dans U , et Lp

loc(U ;V ) pour préciser une intégrabilité ou des bornesessentielles locales.

– On dit qu’une suite (En) de sous-ensembles de RN converge vers E dans

L1(RN) si 1En → 1E dans L1(RN) lorsque n→ +∞.

Short introduction

The subject of this work is the comprehension of some problems related to frontpropagation phenomena. We place ourselves in the framework of viscosity solu-tions, which enables one to define non smooth evolutions, in the very frequent caseswhere a classical evolution does not exist.

The concern of Chapter 1 is to understand certain regularity properties of thesolutions of the most famous of local geometric equations, the eikonal equation. Weare especially interested in finding perimeter estimates for these solutions, in thecase of a velocity which may change sign.

The rest of this dissertation is devoted to nonlocal equations; the major difficultyassociated to these equations is the absence of inclusion principle, which meansthat two fronts may be well ordered at the initial time but eventually overlap.In this work, we have in mind both the general study of these equations and theparticular comprehension of two models which are rich in difficulties, the dynamicsof dislocations and a FitzHugh-Nagumo type system. Because of the absence ofinclusion principle, both existence and uniqueness of classical viscosity solutions areopen questions in general. To overcome this difficulty, we draw our attention to anotion of weak solution for these equations. In Chapter 2, we establish a generalresult of existence of such solutions, which we apply to our two models. Chapter 3 isdevoted to the study of a particular mode of the dynamics of a dislocation with anadditional mean curvature term. The regularizing effect of this term enables us touse the method of minimizing movements to construct weak solutions of a particularform, closer from a classical solution than those provided by the existence result ofChapter 2. In Chapter 4, we prove uniqueness of a weak solution for the two modelequations, in the case of positive velocities. Finally in Chapter 5, we are interestedin the numerical approximation of these nonlocal front propagation equations, andin particular in the convergence of abstract schemes to a weak solution. Explicitexamples of such schemes are also analyzed for both models.

Let us start with a quick presentation of the difficulties and results obtained.

Short introduction

1 Front propagation

By front, we mean the boundary Γ of a closed subset K ⊂ RN (N ∈ N

∗), andwe talk about propagation when this front depends on a time parameter. We thendenote Γ(t) = ∂K(t) the front at time t, for t ∈ [0, T ] where T > 0 is a final time.We are interested in propagation laws where, in the case of a smooth evolution, thenormal velocity Vx,t of Γ(t) at point x is a function of:

– the point x and time t,– the unit outer normal vector νx,t to K(t) at point x,– the curvature matrix Ax,t = [− ∂νi

∂xj(x, t)]1≤i,j≤N of K(t) at point x and time t

(negative for convex sets),– possibly, the family K(s)s∈[0,T ]. In this case, the equation is said to be

nonlocal.We therefore write our evolution law in the following form:

Vx,t = h[K](x, t, νx,t, Ax,t), (1.1)

where h[K] is defined on RN × [0, T ] × SN−1 × SN , the notation SN−1 standing forthe unit sphere of RN , and SN denoting the set of real square symmetric matricesof size N . If the equation is local, we simply write h(x, t, νx,t, Ax,t).

In the level-set approach to front propagation (see [64, 46]), the front is seen asthe 0-level set of a function u : RN × [0, T ] → R, that is,

K(t) = x ∈ RN ; u(x, t) ≥ 0, Γ(t) = x ∈ R

N ; u(x, t) = 0.In this setting, the level-set equation associated to (1.1) is

ut = h[K]

(x, t,− Du

|Du| ,1

|Du|

(IN − Du⊗Du

|Du|2)D2u

)|Du|, (1.2)

where ut, Du and D2u respectively denote the time derivative, space gradient andspace Hessian matrix of u, ⊗ is the tensor product between vectors of RN , and | · |is the standard Euclidean norm.

We are interested in the class of nonlocal equations for which it is possible totranspose the problem entirely in terms of functions and to get rid of all geometricdata such as K, by writing the previous equation in the form

ut(x, t) = H [1u≥0](x, t,Du(x, t), D2u(x, t)), (1.3)

where 1A denotes the indicator function of a set A:

1A(x) =

1 if x ∈ A,

0 if x /∈ A.

In equation (1.3), the Hamiltonian H [χ] is defined for any χ ∈ L∞(RN × [0, T ]) suchthat 0 ≤ χ ≤ 1, and, if χ is an indicator function, it satisfies

H [χ](x, t, p, A) = h[χ = 1](x, t,− p

|p| ,1

|p|

(IN − p⊗ p

|p|2)A

)|p|

32

Short introduction

for all (x, t, p, A) ∈ RN × [0, T ] × R

N \ 0 × SN .

The level-set method consists in achieving the three following goals, so as todefine a front even if Γ(0) is not smooth, or after the onset of singularities:

1. Find u0 : RN → R such that

K(0) = x ∈ RN ; u0(x) ≥ 0 and Γ(0) = x ∈ R

N ; u0(x) = 0.2. Solve in the viscosity sense the problemut(x, t) = H [1u≥0](x, t,Du(x, t), D

2u(x, t)) for (x, t) ∈ RN × (0, T ),

u(x, 0) = u0(x) forx ∈ RN .(1.4)

3. Set

K(t) = x ∈ RN ; u(x, t) ≥ 0 and Γ(t) = x ∈ R

N ; u(x, t) = 0.

For nonlocal equations, the notation H [1u≥0] is justified by the fact that forany indicator function 1u≥0, or more generally for any fixed χ ∈ L∞(RN × [0, T ])satisfying 0 ≤ χ ≤ 1, H [χ] defines a Hamiltionian for which it is possible to solvethe equation

ut(x, t) = H [χ](x, t,Du(x, t), D2u(x, t)), (1.5)

which is a local equation. The resolution of (1.4) then corresponds to the case whereχ is linked to the solution u of (1.5) with initial datum u0 by the relation χ = 1u≥0.

Let us now briefly present the contents of this dissertation.

2 Integral formulations of the eikonal equation

Let us start by presenting the results of Chapter 1 on the local eikonal equation.In this work we aim at studying of the regularity of the solutions of the eikonal equa-tion Vx,t = c(x, t), and more precisely their perimeter. In addition to an independentinterest, it is useful, in certain front propagation problems, to understand the behav-ior of the solutions of the eikonal equation, because this equation appears naturallywhen one “freezes” the nonlocal term, typically in fixed point type arguments: this iswhat (1.5) suggests. The works of Alvarez, Cardaliaguet and Monneau [2] or Barlesand Ley [20] on dislocation dynamics (see below) show in particular that perimeterestimates are one of the keys of the problem. Such estimates exist in the case ofpositive velocities (see [2, 28]): they are based on optimal control arguments whichfail in the case where the velocity changes sign. The goal of Chapter 1 is to studythis case. The starting point of this work is Hadamard’s formula, which enables todifferentiate integrals on evolving domains. More precisely, let Γ(t) = ∂K(t), fort ∈ [0, T ], be a regular and uniformly bounded front evolving with normal velocityVx,t. For any φ ∈ C1(RN × [0, T ]; R), Hadamard’s formula states that

d

dt

∫

K(t)

φ(x, t) dx =

∫

K(t)

φt(x, t) dx+

∫

∂K(t)

Vx,t φ(x, t) dHN−1(x). (2.1)

33

Short introduction

In this formula, we notice the usual integral of the time derivative of the test functionon the domain, as well as an additional boundary term involving the normal velocityof the front. The aim of Chapter 1 is to prove such a formula for the solutions ofthe eikonal equation

Vx,t = c(x, t),

in the framework of viscosity solutions, and in particular without a priori regularityon K(t). Our goal, in fact, is to obtain regularity onK(t). Indeed, applying formallyHadamard’s formula (2.1) to φ = sgn(c), the sign function of c, yields an estimationof the term ∫ T

0

∫

∂K(t)

|c(x, t)| dHN−1(x)dt,

which is a time-integral estimate related to the perimeter of the front ∂K(t): this isindeed a regularity result for the solutions of the eikonal equation.

Results obtained

In Chapter 1, we achieve the following points:

1. Prove integral formulations for the eikonal equation Vx,t = c(x, t), inspired byHadamard’s formula. We will give a formulation for the notion of subsolution (seeTheorem 3.1 page 53) as well as a formulation for the notion of supersolution (seeTheorem 6.1 page 76).

2. Prove that these formulations respectively characterize the notions of subsolutionand supersolution of this equation: see Theorems 4.1 page 60 and 6.3 page 76.

3. Prove that the fronts that are solutions of eikonal equations have locally finiteperimeter in the set c 6= 0 for almost all time, no matter the (compact) initialcondition and the sign of c.

4. Provide an explicit bound related to the perimeter of the K(t)’s. The preciseresults of points 3 and 4 are the subject of Theorem 7.1 page 77.

3 Nonlocal equations without comparison principle

In the rest of this work, we are interested in evolution laws for which the normalvelocity Vx,t of ∂K(t) at point x is a function of the set K(t) itself, and even possiblyof all sets K(s) for s ∈ [0, T ]. These equations are said to be nonlocal, in contrastwith the case where Vx,t only depends on the data at x, t.

The main difficulty linked with nonlocal equations is the absence of comparisonprinciple for the corresponding level-set equation, which means that if u is a sub-solution and v is a supersolution of ut = H [1u≥0](x, t,Du,D

2u) in RN × (0, T ),

then

u∗(x, 0) ≤ v∗(x, 0) in RN does not imply u∗(x, t) ≤ v∗(x, t) in R

N × [0, T ).

34

Short introduction

Indeed, asking for a comparison principle amounts to requiring, χ1 and χ2 beingfixed in L∞(RN × [0, T ]; [0, 1]), that

χ1 ≤ χ2 ⇒ H [χ1] ≤ H [χ2].

This does not hold in general. To illustrate this point we now present two modelswhich we will study in details throughout this dissertation.

3.1 Two nonlocal models

3.1.1 Dislocation dynamics

A dislocation is a linear defect corresponding to a discontinuity in the crystallineorganisation of metals. These lines move in prescribed planes and are the majormicroscopic explanation of plastic deformations observed in crystals at the macro-scopic scale. We refer to [4, 66] and the dissertation of Forcadel [42] for a detailedpresentation of this model. In the model of [66], the normal velocity of a dislocationline ∂K(t) is given by the following nonlocal law:

Vx,t = c0(·, t) ⋆ 1K(t)(x) + c1(x, t)

=

∫

K(t)

c0(x− y, t) dy + c1(x, t),(3.1)

of associated level-set equationut(x, t) =

[c0(·, t) ⋆ 1u(·,t)≥0(x) + c1(x, t)

]|Du(x, t)| in RN × (0, T ),

u(x, 0) = u0 in RN .(3.2)

The kernel c0, which gives the nonlocal term by space convolution with 1K(t), is asymmetric function depending on the material; the term c0(·, t)⋆1K(t) represents theforce coming from the elastic field created by the dislocation, and therefore the forceapplied by the dislocation on itself. The term c1 represents external constraints.The kernel c0 has zero mean value, and in particular changes sign; as a consequencethe comparison principle is not satisfied in this model: the relation χ1 ≤ χ2 doesnot imply c0(·, t) ⋆ χ1(·, t) ≤ c0(·, t) ⋆ χ2(·, t) for any t ∈ [0, T ].

Another possible model for the dynamics of dislocations, which takes into accountthe line tension of the dislocation, consists in adding, in the expression of the velocityVx,t, the mean curvature of ∂K(t) at x,

Hx,t = Tr(Ax,t) = −N∑

i=1

∂νi

∂xi(x, t)

(recall that Ax,t denotes the curvature matrix, and ν(x, t) the outer unit normalvector, of K(t) at x). The resulting geometric equation is

Vx,t = Hx,t + c0(·, t) ⋆ 1K(t)(x) + c1(x, t), (3.3)

35

Short introduction

and its associated level-set equation isut =

[div(

Du|Du|

)+ c0(·, t) ⋆ 1u(·,t)≥0(x) + c1(x, t)

]|Du| in RN × (0, T ),

u(·, 0) = u0 in RN .(3.4)

This is also a nonlocal equation which does not satisfy a comparison principle.

3.1.2 A FitzHugh-Nagumo type system

We will also closely study the following system,

ut = α(v)|Du| in RN × (0, T ),

vt − ∆v = g+(v)1u≥0 + g−(v)(1 − 1u≥0) in RN × (0, T ),

u(·, 0) = u0, v(·, 0) = v0 in RN ,

(3.5)

obtained as the limit as ε→ 0 of the FitzHugh-Nagumo system appearing in neuralwave propagation or in chemical kinetics:

uε

t − ε∆uε = ε−1f(uε, vε),


in RN × (0, T ), where, for (u, v) ∈ R2,f(u, v) = u(1 − u)(u− a) − v (0 < a < 1),

g(u, v) = u− γv (γ > 0).

The functions α, g+ and g− appearing in (3.5) are linked with f and g (see [48, 72]).

System (3.5) is a coupled system, the first equation of which describes the move-ment of a front Γ(t) = u(·, t) = 0 evolving with velocity α(v), v being itselflinked to Γ(t) as the solution of a reaction-diffusion equation whose source dependson the interface Γ(t). However, this system can be put in the form (1.4): forχ ∈ L∞(RN × [0, T ]; [0, 1]), let v be the solution of

vt − ∆v = g+(v)χ+ g−(v)(1 − χ) in RN × (0, T ),

v(·, 0) = v0 in RN ,(3.6)

and let c[χ](x, t) = α(v(x, t)). Then (3.5) is equivalent tout(x, t) = c[1u≥0](x, t)|Du(x, t)| in RN × (0, T ),

u(x, 0) = u0 in RN .(3.7)

We remark that in this model the solution v of (3.6), and therefore the velocityc[χ], depends on χ in a nonlocal way in both space and time: indeed the explicitresolution of the heat equation (3.6) shows that for all (x, t) ∈ RN × (0, T ),

v(x, t) =

∫

RN


+

∫ t

0

∫

RN


(3.8)

36

Short introduction

where G is the Green function defined by

G(y, s) =1

(4πs)N/2e−

|y|2

4s . (3.9)

Besides, since the function α is not non-decreasing, this system does not satisfy acomparison principle.

3.2 Weak solutions of nonlocal equations

Because of the absence of comparison principle, the classical techniques for build-ing a viscosity solution to (1.4) fail. A possibility to overcome this difficulty consistsin adapting the notion of solution, by proposing a notion of weak solution of (1.4).Let us now present this notion, which is central in this work. It uses the notion ofL1-viscosity solution, which is an adaptation of the notion of viscosity solution in thecase where the Hamiltonian is only measurable in time. We refer to [50, 61, 62, 27, 26]for a complete presentation of the theory of L1-viscosity solutions.

Definition 3.1 (Weak solution).Let u : RN × [0, T ] → R be a continuous function. We say that u is a weak

solution of (1.4) if there exists χ ∈ L∞(RN × [0, T ]; [0, 1]) such that

1. u is a L1-viscosity solution ofut(x, t) = H [χ](x, t,Du,D2u) in R

N × (0, T ),

u(·, 0) = u0 in RN .(3.10)

2. For almost all t ∈ [0, T ],

1u(·,t)>0 ≤ χ(·, t) ≤ 1u(·,t)≥0 a.e. in RN .

Moreover, we say that u is a classical viscosity solution of (1.4) if in addition, foralmost all t ∈ [0, T ],

1u(·,t)>0 = 1u(·,t)≥0 a.e. in RN .

4 Existence of weak solutions for general nonlocal

equations

In Chapter 2, we start by considering the question of existence of such weaksolutions to nonlocal equations (1.4). The following method is not new: it alreadyappears in [48] for the FitzHugh-Nagumo system. We have used it to obtain resultsof existence of weak solutions for general nonlocal equations. Consider the set-valuedmapping

ξ : χ ∈ L∞(RN × [0, T ]; [0, 1])

7→ χ′; 1u(·,t)>0 ≤ χ′(·, t) ≤ 1u(·,t)≥0 for almost every t ∈ [0, T ],

where u is the L1-viscosity solution of ut = H [χ](x, t,Du,D2u)

with initial value u(·, 0) = u0.

37

Short introduction

If ξ has a fixed point χ, in the sense that χ ∈ ξ(χ), then the corresponding func-tion u is a weak solution of (1.4). Therefore, we can prove the existence of weaksolutions to (1.4) as a consequence of Kakutani’s fixed point theorem for set-valuedmappings. The main assumption we have to check in order to use this theorem isthe upper semicontinuity of the set-valued mapping ξ for the topology of the weak-∗convergence on L∞(RN × [0, T ]; [0, 1]), which means that

χn

n→∞χ in L∞ weakly-∗

χ′n ∈ ξ(χn)

n→∞χ′ in L∞ weakly-∗ ⇒ χ′ ∈ ξ(χ).

In our context, it is a direct consequence of the new stability result of Barles [15],which ensures stability for L1-viscosity solutions under the following minimal as-sumption:

(H) :χn n→∞

χ in L∞ weakly-∗ ⇒∫ t

0

H [χn](x, s, p, A)ds −→n→∞

∫ t

0


locally uniformly with respect to x, t, p, A.

Results obtained

1. If for any fixed function χ, the HamiltonianH [χ] satisfies some regularity assump-tions – which guarantee in particular a comparison principle for the local equationut = H [χ](x, t,Du,D2u) – and if the mapping χ 7→ H [χ] satisfies the continuityassumption (H), then there exists at least one weak solution to (1.4). In fact, wegive two different existence results: the first one corresponds to the favorable casewhere H [χ] is defined for any function χ, and in particular without restriction onthe size of its support; this is Theorem 4.1 page 85. The second result (Theorem5.1 page 92) applies to equations for which it is necessary to impose a restriction onχ for H [χ] to be defined, such as the boundedness of its support: for instance, it isthe case of equations involving the volume of the set K(t). In both cases, the resultapplies to very general equations: these equations need not be geometric (even ifin applications, we have in mind front propagation models, hence geometric equa-tions), and the Hamiltonian H [χ] may depend on the values of χ(·, s) for any times ∈ [0, T ], past or future.

2. As an application of the first existence result, we prove existence of weak solutionsfor the dislocation dynamics equation (3.2) and the FitzHugh-Nagumo problem(3.7). These results were already known (cf. [16] and [48, 72]) but we easily recoverthem from our general theorem. As an application of the second result, we deal withequations depending on the volume of K(t).

5 Dislocation dynamics with a mean curvature term

We would also easily obtain existence of weak solutions for the dislocation dy-namics equation with an additional mean curvature term (3.4) as an application of

38

Short introduction

the general existence result (Theorem 4.1 page 85). However, the interest of themean curvature term in this model, beyond the fact that it takes into account theline tension of the dislocation, is its regularizing aspect, so that we expect to obtainbetter weak solutions than those described in Definition 3.1. This is the subject ofChapter 3 that we now present.

In this chapter, we build a weak solution of (3.4) by constructing a particularfunction χ(·, t), not having the ideal form 1u(·,t)≥0, but nevertheless under theform of an indicator function 1E(t), the mapping t 7→ 1E(t) having some regularityproperties.

We build E as a minimizing movement for the evolution law (3.3): thisconsists in finding E as the flow which minimizes the energy associated to this law.This idea is due to Almgren, Taylor and Wang [1], and independently to Luckhausand Sturzenhecker [55], for the mean curvature motion.

The minimizing movements approach amounts to discretizing the evolution law

Vx,t = Hx,t + c0(·, t) ⋆ 1K(t)(x) + c1(x, t) (5.1)

in time: if h is a time step, we look for a sequence of sets Eh(k), for all k ∈ N withkh ≤ T , such that for all x ∈ ∂Eh(k + 1),

±d∂Eh(k)(x)

h= Hx,(k+1)h + c0(·, (k + 1)h) ⋆ 1Eh(k+1)(x) + c1(x, (k + 1)h), (5.2)

where we choose the + sign if x /∈ Eh(k), and the − sign otherwise; Hx,(k+1)h

denotes the mean curvature of Eh(k+ 1) at x. This corresponds to an implicit timediscretization of (5.1).

Setting Eh(0) = E0 for some compact initial datum E0 having finite perimeter,we then build Eh(k + 1) from Eh(k) by seeing (5.2) as the formal Euler-Lagrangeequation associated to the minimization of the functional

E 7→ F(h, k + 1, E, Eh(k))

= P (E) +1

h

∫

E∆Eh(k)

d∂Eh(k)(x) dx

−∫

E

(1

2c0(·, (k + 1)h) ⋆ 1E(x) + c1(x, (k + 1)h)

)dx,

which contains the energy of the system as well as a penalization term correspondingto the discrete velocity. In this term, for E,F ⊂ RN , E∆F = (E ∪ F ) \ (E ∩ F ) isthe symmetric difference between E and F .

We expect any minimizer of this functional to be a solution of (5.2). The idea isthen to pass Eh to the limit as h→ 0 by choosing, for any fixed t ∈ [0, T ], k = [t/h],so that kh → t as h → 0. We obtain a minimizing movement t 7→ E(t) associatedto the functional F with initial condition E0.

The results obtained, which are the subject of Chapter 3, are the following:

39

Short introduction

Results obtained

Under adapted assumptions on c0 and c1, we have:

1. If LN(E0) = 0, then there exists Hölder continuous minimizing movements as-sociated to F with initial condition E0, which means that there exists at least oneminimizing movement E and a constant C > 0 such that for all t, s ∈ [0, T ],

|E(t) ∆E(s)|L1(RN ) ≤ C|t− s| 1N+1 .

This result corresponds to Theorem 1.3 page 100.

2. The corresponding solution ofut =

[div(

Du|Du|

)+ c0(·, t) ⋆ 1E(t)(x) + c1(x, t)

]|Du| in RN × (0, T ),

u(·, 0) = u0 in RN ,

is a weak solution of (3.4) with χ = 1E: this is Theorem 1.4 page 101. This resultstrongly uses the Euler-Lagrange equation (5.2), which itself is obtained by theregularity theory for F -minimizers [49]. It is inspired by the work of Chambolle[33] on the consistency between the notions of minimizing movement and viscositysolution for the mean curvature motion.

3. If E0 is a compact set with C3+α boundary, then there exists t0 > 0 and aunique smooth evolution Er(t)0≤t≤t0 such that ∂Er(t) is of class C3+α for any t,Er(0) = E0, and with normal velocity given by (3.3): see Theorem 1.6 page 101.

Moreover, any minimizing movement E associated to F with initial conditionE0 satisfies E(t) = Er(t) for all t ∈ [0, t0] and almost everywhere in RN : this is theobject of Theorem 1.5 page 101.

6 Uniqueness of a weak solution for the two model

problems

As we have just seen, the results of Chapters 2 and 3 are concerned with theexistence of weak solutions to nonlocal equations of the form (1.4). We also naturallyconsidered the problem of uniqueness in the case of the models presented above, thedynamics of dislocations (3.2) and the FitzHugh-Nagumo system (3.5) equivalent to(3.7). This is the problem addressed in Chapter 4, that we now expose.

Both equations (3.2) and (3.7) are first-order particular cases of (1.4), havingthe form of a nonlocal eikonal equation

ut(x, t) = c[1u≥0](x, t) |Du(x, t)| in RN × (0, T ),

u(·, 0) = u0 in RN .(6.1)

The question of uniqueness is difficult: it is linked with the fattening phenomenon,which can happen for first-order equations as soon as the velocity changes sign.

40

Short introduction

Barles, Cardaliaguet, Ley and Monneau [16] built an explicit counter-example inthe case of the dislocation equation. For the general equation (6.1), when on thecontrary there exists δ > 0 such that for all χ ∈ L∞(RN × [0, T ]; [0, 1]), we havec[χ] ≥ δ, it is then known (see Ley [53]) that any weak solution u is in fact a classicalsolution, because the level-set u = 0 has zero measure: the fattening phenomenondoes not happen. The question of existence of classical solutions therefore hasan affirmative answer. However, this positivity of the velocity does not replace acomparison principle, and the question of uniqueness remains.

Nevertheless, techniques especially adapted to positive velocities enabled Al-varez, Cardaliaguet and Monneau [2] and Barles, Ley [20] to prove existence anduniqueness for the dislocation equation (by different methods). Their results aresynthesized and improved in [16]. In the articles [2] and [20], uniqueness is obtainedby contracting fixed point methods, based on perimeter estimates on the level-setsu(·, t) = α of the solutions. These estimates come from bounds on the curvatureof these sets, corresponding to an interior ball condition on the sets u(·, t) ≥ α,which demands a rather strong regularity in space for c[χ] (typically C1,1). As far asthe FitzHugh-Nagumo system (3.5) is concerned, uniqueness was up to now an openquestion, even in the case of a positive velocity. In addition, for this system, thevelocities c[χ] do not have the C1,1 regularity and the techniques of [2, 20, 16] cannot be applied directly. The aim of Chapter 4 is to prove uniqueness of a classicalsolution for these two equations, under the assumption that the velocity is positive.For the dislocation dynamics model, the improvement over previous results is thelower regularity required from the velocities.

For the dynamics of dislocations, when the kernel c0 is bounded, estimates inL1([0, T ]) on the volume of sets of the type a ≤ u(·, t) ≤ b for the solution u of theeikonal equation are sufficient to conclude. Besides, they are rather easy to obtaindirectly.

On the contrary, for the FitzHugh-Nagumo system, the Green kernel defined by(3.9) is not bounded but integrable, and L1([0, T ]) type estimates are not sufficient;instead we need L∞([0, T ]) estimates of the perimeter of sets like u(·, t) = α.To obtain such estimates, we show that these sets statisfy, if not an interior ballcondition, an interior cone condition, from which we deduce explicit estimates ontheir perimeter.

Results obtained

Under assumptions (H1) and (H2), we obtained the following results:

1. Equations (3.2) and (3.7) have a unique weak solution, which is a classical solution:see Theorems 3.1 page 138 and 4.1 page 141. The proofs of these results use thefollowing auxiliary results, of independent interest:

2. An L1([0, T ]) estimate on the volume of sets like a ≤ u(·, t) ≤ b for the solutionu of the eikonal equation: see Proposition 5.5 page 148.

3. The solutions of eikonal equations develop an interior cone for all times t ∈ [0, T ].

41

Short introduction

This result is the object of Theorem 5.9 page 152.

4. A compact set K having the interior cone property satisfies

HN−1(∂K) < +∞with a precise estimation of this measure: see Theorem 5.8 page 149.

7 Convergence of approximation schemes for nonlo-

cal equations

In Chapter 5, we are interested in the approximation of nonlocal equations of thetype (1.4). A lot of work has been done on this question. For example, Cardaliaguetand Pasquignon [32] and Slepčev [70] have studied the case of equations satisfyingan inclusion principle. Alvarez, Carlini, Monneau and Rouy [3] have proposed ascheme for the dynamics of dislocations, which does not satisfy this principle. FastMarching methods are also widely exploited. On this question, one can find in thebooks by Sethian [68] or Osher and Fedkiw [63], numerous schemes for a large rangeof applications. Our work is motivated by the existence results obtained in Chapter2, as well as the result of convergence of approximation schemes of Barles andSouganidis [23] for local equations. For our part, we want to prove the convergenceof general schemes to a – or in the favorable cases to the – weak solution of thenonlocal equation (1.4).

Let h = T/n be a time step, for some n ∈ N∗, and let

Πh =⋃

(i1,...,iN )∈ZN

xi1,...,iN =⋃

(i1,...,iN )∈ZN

(i1∆1, . . . , iN∆N)

be an associated space grid with space steps ∆1, . . . ,∆N . For x = (x1, . . . , xN ) ∈ RN ,let

xh := ([x1/∆1 + 1/2]∆1, . . . , [xN/∆N + 1/2]∆N) ∈ Πh

denote its projection on this grid, so that if x ∈ Qi1,...,iN , then xh = xi1,...,iN .

For any x ∈ Πh, kh ≤ T , u : Πh → R and any χ : Πh × [0, T ] → [0, 1] withbounded support, we define a discrete Hamiltonian Hh[χ](x, kh, u), which is anapproximation of H [χ](x, t,Du(x), D2u(x)), depending on the values

χ(xi1,...,iN , lh) for (i1, . . . , iN) ∈ ZN and 0 ≤ l ≤ k,

u(xi1,...,iN , kh) for (i1, . . . , iN) ∈ ZN .

In particular, we also consider equations depending on the past, such as the FitzHugh-Nagumo problem (3.7).

The approximation schemes that we have in mind have the following form: forall k ∈ N satisfying (k + 1)h ≤ T , and all x ∈ Πh, we set


(7.1)

42

Short introduction

We then extend uh to a piecewise constant function on RN × [0, T ] by setting for

any (x, t),uh(x, t) = uh(xh, [t/h]h).

In the spirit of the result of Barles and Souganidis, we consider conditionallymonotone, stable and consistent schemes. The conditional monotonicity meansthat for any fixed function χ : Πh × [0, T ] → [0, 1] and k ∈ N such that kh ≤ T ,

u ≤ v ⇒ u(x) +H [χ](x, kh, u) ≤ v(x) +H [χ](x, kh, v) for all x ∈ Πh.

The stability of the scheme corresponds to uniform L∞ bounds on the solutions of(7.1). Let us stress the fact that these two properties concern the scheme with afixed χ, that is to say, a scheme corresponding to the local equation (3.10): we donot require the monotonicity of the nonlocal scheme (7.1).

Our result of convergence is based on the idea of proof of the existence theorem4.1 of Chapter 2: indeed, the sequence (1uh≥0) is bounded in L∞(RN × [0, T ]), sothat there exists a subsequence, also indexed by h for simplicity, converging to acertain χ ∈ L∞(RN × [0, T ]; [0, 1]) for the topology of the weak-∗ convergence. Ourgoal is then to show that (uh) converges locally uniformly to a weak solution of (1.4).Of course, this convergence depends on the good choice of the discrete HamiltonianHh, that is to say, its consistency with respect to H . To get the intuition of theright consistency condition to impose, it is natural to look at stability results forequation (3.10), since they give conditions on the convergence of the Hamiltonianswhich guarantee convergence of the corresponding solutions. But, as we have alreadypointed out, in our framework the best stability result is the one by Barles citedabove (see [15]). The proof of convergence for our schemes is therefore naturallyinspired from the proof of this stability result. The main assumption of this theorem(see (H) page 38) becomes, in our discrete setting:

χh h→0

χ in L∞ weakly-∗ ⇒[t/h]−1∑

l=0

Hh[χh](xh, s, p, A)ds −→h→0

∫ t

0


locally uniformly in x, t, p, A; this is our consistency assumption.

Results obtained

1. Under this consistency assumption, and the assumptions of conditional mono-tonicity and stability, we prove the convergence of general schemes of the form (7.1)to a weak solution of (1.4): see Theorem 2.2 page 165.

2. In the particular cases of the dislocation dynamics equation (3.2) and FitzHugh-Nagumo system (3.5), we also give explicit examples of schemes and prove theirconvergence according to our general result: this corresponds to Theorems 3.1 page171 and 3.2 page 176. The scheme that we consider for the dynamics of dislocationsis implemented in [3].

43

Chapitre 1

Integral formulations of the

geometric eikonal equation

Ce chapitre est issu de l’article [58] publié dans Interfaces and Free Boundaries.

On y démontre des formulations intégrales de l’équation eikonale ut = c(x, t)|Du|,équivalentes à la notion de solution de viscosité. On utilise ensuite ces formula-tions intégrales pour l’étude de la régularité du front: sous certaines hypothèses derégularité sur la vitesse c, on montre que le front est de périmètre localement finidans l’ensemble c 6= 0, et on donne une estimation de type L1 en temps de sonpérimètre.

Chapitre 1. Integral formulations of the geometric eikonal equation

Abstract

We prove integral formulations of the eikonal equation ut = c(x, t)|Du|,equivalent to the notion of viscosity solution in the framework of the

set-theoretic approach to front propagation problems. We apply these

integral formulations to investigate the regularity of the front: we prove

that under regularity assumptions on the velocity c, the front has locally

finite perimeter in c 6= 0, and we give a time-integral estimate of its

perimeter.

Key words and phrases: Front propagations, eikonal equation,viscosity solutions, set-theoretic approach, functions of boundedvariation and sets of finite perimeter.

1 Introduction

We are interested in generalized time evolutions of subsets K(t) of RN , N ≥ 1,governed by the following geometric law:

Vx,t = c(x, t), (1.1)

where Vx,t denotes the normal velocity of ∂K(t) at a point x. If for example K(t)can be represented by

K(t) = x ∈ RN ; u(x, t) ≥ 0 , ∂K(t) = x ∈ R

N ; u(x, t) = 0

for some C1 function u : RN × [0, T ] → R such that u(x, t) = 0 implies that thespace gradient Du(x, t) 6= 0, then a classical calculation yields

Vx,t =ut(x, t)

|Du(x, t)| ,

where ut denotes the time derivative of u, and | · | the usual euclidian norm on RN

(|x| = 〈x, x〉 12 ), so that u satisfies the so-called eikonal equation

ut = c(x, t)|Du| (1.2)

for all (x, t) ∈ RN × (0, T ) such that u(x, t) = 0. If the front ∂K(t) is not assumedto be regular, equation (1.2) gives a generalized way of studying the evolution, andthe notion of viscosity solution of this equation provides a satisfactory framework todo so. More precisely, one possible generalized solution of the geometric evolution(1.1), called a set-theoretic solution, is a family K(t) of subsets of RN such that(x, t) 7→ 1K(t)(x) is a discontinuous viscosity solution of (1.2), where 1E is theindicator function of a set E. We refer to [36] for a complete overview of viscositysolutions, and to [46] for details about set-theoretic solutions. Various applicationsof the notion of viscosity solutions can also be found in [12, 13, 14, 37, 54], to mentionbut a few. Other approaches to the problems of propagating fronts are presented in[7, 24, 71].

46

1. Introduction

Our work is motivated by a model for the dynamics of dislocation lines in acrystal, which gives rise to a Hamilton-Jacobi equation with a nonlocal term (see(3.1)-(3.2) in the general introduction). In questions related to existence and unique-ness for this nonlocal equation, it is interesting to understand the behavior of thesolutions of the eikonal equation, and especially the regularity properties of the front∂K(t). This has been done for example by Evans and Spruck [40] for the motion bymean curvature. In the case of the geometric eikonal equation, the regularity of thefront has recently been investigated for positive velocities by Alvarez, Cardaliaguetand Monneau [2], and using another method by Barles and Ley [20]; their worksshow that a good property of regularity to investigate is that of finite perimeter.More precisely, it is proved in [2] that the K(t)’s have finite perimeter if c > 0 andK(0) has the interior ball property, i.e. is the union of closed balls of some fixed ra-dius. Cardaliaguet and Cannarsa [28] then improved and generalized the estimatesof [2]. However, results in this spirit lack for velocities with no sign.

To achieve such results, we search for analytic ways of studying the geometricequation (1.1), and we first focus on the case of set-theoretic subsolutions. One cluein this direction is Hadamard’s formula in the case where K(t) is open, bounded,and K(t) evolves smoothly in time: for all φ ∈ C1(RN × [0, T ]; R),

d

dt

∫

K(t)

φ(x, t) dx =

∫

K(t)

φt(x, t) dx+

∫

∂K(t)

Vx,t φ(x, t) dHN−1(x),

where HN−1 denotes the (N − 1)-dimensional Hausdorff measure. In particular ifVx,t ≤ c(x, t) in the classical sense, then for all φ ∈ C1(RN × [0, T ]; R+), we have

d

dt

∫

K(t)

φ(x, t) dx ≤∫

K(t)

φt(x, t) dx+

∫

∂K(t)

c(x, t)φ(x, t) dHN−1(x). (1.3)

The problem is that the term∫

∂K(t)c(x, t)φ(x, t) dHN−1(x) does not make sense in

the viscosity solution framework: the regularity of K(t) in terms of perimeter isunknown (actually, one of the reasons to consider this formula is to give a meaningto this term). However, if we set

Kε(t) = x ∈ RN ; dK(t)(x) < ε,

where dF denotes the distance function to any closed non-empty set F ⊂ RN ,then Kε(t) has finite perimeter and one of our main results is that (1.3) admitsthe following integral generalization: under standard regularity assumptions on thevelocity c : RN × [0, T ] → R, we have for all t1 and t2 satisfying 0 ≤ t1 ≤ t2 ≤ T ,for almost all ε > 0, and for all φ ∈ C1(RN × [0, T ]; R+),[∫

Kε(t)

φ(x, t) dx

]t2

t1

≤∫ t2

t1

∫

Kε(t)

φt(x, t) dxdt+

∫ t2

t1

∫

dK(t)=εcε(x, t)φ(x, t) dHN−1(x)dt,

where we have introduced a perturbation of the velocity by setting

cε(x, t) = max|y−x|≤ε

c(y, t).

47


We find in [2], in the case where c > 0, an integral formulation valid for anyfamily K(t) such that 1K(t) satisfies the eikonal equation almost everywhere. Inour case, although the perturbation by ε seems restrictive at first glance, we have incompensation considerable freedom in choosing ε. As a consequence, we will provethat our integral formulation is actually equivalent to the notion of set-theoreticsubsolution of the eikonal equation.

After setting the notation in Section 2, we prove the integral formulation of theeikonal equation for subsolutions (Theorem 3.1), in Section 3. Conversely, we provein Section 4 that this integral formulation characterizes the notion of set-theoreticsubsolution of (1.1): this is Theorem 4.1. Then in Section 5 we use the integral for-mulation for subsolutions to investigate regularity properties of the K(t)’s and givea time-integral estimate of their perimeter. These results are contained in Theorem5.1. All the results can be adapted to the notion of set-theoretic supersolutions, asexpected. We collect the corresponding results in Section 6. Finally in Section 7,under a regularity assumption on the evolution related to the non-empty interiordifficulty arising in the level-set method, we combine these results for sub- and su-persolutions to prove in Theorem 7.1 that for almost all t ∈ [0, T ], K(t) has locallyfinite perimeter in c(·, t) 6= 0, and that we have the following estimate:

∫ T

0

∫

∂∗K(t)

|c(x, t)| dHN−1(x)dt

≤[∫

K(t)

sgn(c(x, t)) dx

]T

0

+ 2

∫ T

0

∫

K(t)∩c(·,t)=0

|ct(x, t)||Dc(x, t)| 1ct<0(x, t) dHN−1(x)dt,

where ∂∗K(t) is the reduced boundary of K(t), and sgn(c) the sign of c.

2 Notation and tools

Let us start with standard notation: B(x, r) (resp. B(x, r)) denotes the open(resp. closed) ball of radius r > 0 centered at x ∈ RN . The notation Ec stands forthe complement of a set E, and Lp denotes the Lebesgue measure on Rp. FinallyC0(U ;V ) (resp. C1(U ;V )) is the set of continuous (resp. continuously differentiable)functions from U ⊂ Rp to V ⊂ Rq. We add a subscript c to indicate that inaddition these functions have compact support in U : C0

c (U ;V ), C1c (U ;V ). Finally we

abbreviate “upper semicontinuous” (resp. “lower semicontinuous”) to usc (resp.lsc).

2.1 Viscosity solutions and set-theoretic approach

We first give the results that ensure the existence of set-theoretic sub- and su-persolutions of the eikonal equation with initial value:

ut = c(x, t)|Du| in RN × (0, T ),

u(x, 0) = u0(x) in RN .

(2.1)

48

2. Notation and tools

For the convenience of the reader, we recall the definitions of a viscosity sub-and supersolution of the eikonal equation and of a solution of problem (2.1):

Definition 2.1 (Crandall, Lions [37]).1. We say that u : R

N × [0, T ] → R is a viscosity subsolution (resp. superso-lution) of ut = c(x, t)|Du| in RN × (0, T ) if u is usc (resp. lsc) and if for all testfunction φ of class C1 such that u − φ has a local maximum (resp. minimum) at(x, t) ∈ RN × (0, T ), we have

φt(x, t) ≤ c(x, t)|Dφ(x, t)| (resp. ≥).

2. We say that u : RN × [0, T ] → R is a viscosity solution of (2.1) if u is both aviscosity sub- and supersolution of ut = c(x, t)|Du| in RN ×(0, T ) and if u(·, 0) = u0.

Theorem 2.2 (Crandall, Lions [37]). Suppose that c : RN×[0, T ] → R is continuous,bounded, and Lipschitz continuous with respect to the space variable. Then the prob-lem (2.1) has a unique uniformly continuous viscosity solution u : RN × [0, T ] → R

for any uniformly continuous initial value u0.

Let K(0) be fixed with K(0) 6= ∅, RN , let Ω(0) = int(K(0)), and dK(0) denotethe signed distance function to K(0), namely

dK(0)(x) =

d∂K(0)(x) if x ∈ K(0),

−dK(0)(x) if x /∈ K(0).

Let u be the solution of (2.1) with initial value u0 = dK(0). A technique adaptedfrom Barles, Soner and Souganidis [22, Theorem 2.1], which relies on the stabilitytheorem for viscosity solutions, shows that 1u≥0 is also a subsolution of (1.2),while 1u>0 is a supersolution. Thus if we set K(t) = x ∈ RN ; u(x, t) ≥ 0, andΩ(t) = x ∈ RN ; u(x, t) > 0, we obtain a new subsolution u : (x, t) 7→ 1K(t)(x)and a new supersolution u : (x, t) 7→ 1Ω(t)(x) of (1.2) in RN × (0, T ), with respectiveinitial values 1K(0) and 1Ω(0).

We notice that the upper semicontinuity of u on RN × [0, T ] required in thedefinition of subsolution is equivalent to the fact that the graph of K : t 7→ K(t),i.e. Graph(K) =

⋃t∈[0,T ] t ×K(t), is closed in [0, T ] × RN . Likewise, the lower

semicontinuity of u on RN × [0, T ] is equivalent to the graph of Ω : t 7→ Ω(t) beingopen in [0, T ] × R

N .

From now on we are going to focus on these so-called set-theoretic sub- andsupersolutions of the form u : (x, t) 7→ 1K(t)(x) and u : (x, t) 7→ 1Ω(t)(x) respectively.We first gather some properties of the geometric evolutions t 7→ K(t) and t 7→ Ω(t).We will write Ωc(t) for (Ω(t))c.

Proposition 2.3. Let c satisfy the assumptions of Theorem 2.2, and let L = ‖c‖∞.

1. Let K : [0, T ] → P(RN )\∅ be such that u : (x, t) 7→ 1K(t)(x) is a subsolutionof (1.2) in RN × (0, T ). Then for all t ∈ [0, T ] and for all s ∈ [0, T − t), we have

49


K(t+s) ⊂ K(t)+ B(0, Ls). In particular, the evolution is bounded on [0, T ) if K(0)is compact.

2. Let Ω : [0, T ] → P(RN) \ RN be such that u : (x, t) 7→ 1Ω(t)(x) is asupersolution of (1.2) in RN ×(0, T ). Then for all t ∈ [0, T ] and for all s ∈ [0, T−t),we have Ωc(t+ s) ⊂ Ωc(t) + B(0, Ls).

Proof. Let us prove the first point. Since u : (x, t) 7→ 1K(t)(x) is a viscositysubsolution of ut = c(x, t)|Du|, we deduce from the inequality |c| ≤ L that forany fixed t ∈ [0, T ), u1 : (x, s) 7→ u(x, t + s) is a subsolution of ut = L|Du| inRN × (0, T − t) with u1(·, 0) = 1K(t). The solution of this last equation with ini-tial value u2(·, 0) : x 7→ max(1 − dK(t)(x), 0), which is uniformly continuous, isknown and given by u2 : (x, s) 7→ max|y−x|≤Ls u2(y, 0). Since u1 and u2 satisfyu1(·, 0) ≤ u2(·, 0), we deduce from the comparison principle (see [22, Theorem 1.3])that for all (x, s) ∈ RN × [0, T − t),

u1(x, s) ≤ u2(x, s), i.e. u(x, t+ s) ≤ max|y−x|≤Ls

u2(y, 0).

Therefore if x /∈ K(t) + B(0, Ls), then for all y with |y − x| ≤ Ls, y /∈ K(t), i.e.u2(y, 0) < 1. As a result u(x, t+ s) < 1, that is to say x /∈ K(t+ s).

The proof of the second point follows from the first since it is straightforward tocheck that (x, t) 7→ 1Ωc(t)(x) = 1 − u(x, t) is a subsolution of ut = −c(x, t)|Du| inR

N × (0, T ). However, we point out that we can not deduce from this property onΩc(t) that the evolution t 7→ Ω(t) is bounded on [0, T ) if Ω(0) is bounded.

We now state properties of regularity in time of the distance function to thefronts:

Proposition 2.4. Under the assumptions of Proposition 2.3, (x, t) 7→ dK(t)(x) islsc on RN × [0, T ], and for all x ∈ RN , t 7→ dK(t)(x) is left continuous on (0, T ).The same conclusions hold for (x, t) 7→ dΩc(t)(x).

Proof. We only treat the case of K since the proof for Ω is the same.Step 1. Let us show that (x, t) 7→ dK(t)(x) is lsc on R

N × [0, T ], by first consideringA = (x, t) ∈ RN × [0, T ]; x /∈ K(t), which is open in RN × [0, T ] by assumption.Let (xn, tn) be a sequence of elements of A converging to (x, t) ∈ A. Since K(t) isclosed and non-empty for all t, the distance of xn (resp. x) to K(tn) (resp. K(t)) isattained by a certain yn (resp. y):

dK(tn)(xn) = |xn − yn|,yn ∈ K(tn),

dK(t)(x) = |x− y|,y ∈ K(t).

Let us first consider the case of a finite accumulation point of (dK(tn)(xn)), given bythe limit of (|xnk

− ynk|). The sequences (xnk

) and (|xnk− ynk

|) converge, thus (ynk)

is bounded. Passing to a subsequence if necessary, we can assume that it converges,say to a certain y∞. Therefore, since u : (x, t) 7→ 1K(t)(x) is usc, we must haveu(y∞, t) ≥ lim sup(u(ynk

, tnk)) = 1, which means that y∞ ∈ K(t), and by definition

50

2. Notation and tools

of y, |x − y| ≤ |x − y∞|, i.e. dK(t)(x) ≤ lim dK(tnk)(xnk

). This is also true for aninfinite accumulation point, obviously. Finally we remark that this also holds fora couple (x, t) such that x ∈ K(t), since in this case dK(t)(x) = 0, the minimumpossible value of the distance function.

Step 2. Let us now prove the second part of the assertion: for all x ∈ RN , themap t 7→ dK(t)(x) is left continuous on (0, T ). Let us fix t ∈ (0, T ), ε > 0 and asequence (tn) converging to t from the left. Thanks to Proposition 2.3, we knowthat K(t) ⊂ K(tn) + B(0, L(t− tn)) ⊂ K(tn) + B(0, ε) for tn close enough to t. Lety ∈ K(t) be so that dK(t)(x) = |x−y|. Then for n large enough, y ∈ K(tn)+B(0, ε),so that dK(tn)(x) ≤ dK(t)(x) + ε, and this proves that t 7→ dK(t)(x) is left usc.

2.2 Semiconvex functions and subdifferential of a convex func-

tion

We refer to [29] for the notion of semiconvex function. Recall that a real-valuedfunction f defined on a convex subset Ω of RN is semiconvex with constant M ≥ 0if x 7→ f(x) + M

2|x|2 is convex on Ω (and semiconcave if −f is semiconvex), which

amounts to saying that for all (x, y) ∈ Ω2 and all λ ∈ [0, 1],

f(λx+ (1 − λ)y) ≤ λf(x) + (1 − λ)f(y) +M

2λ(1 − λ)|x− y|2.

For f of class C2 and Ω open, this is also equivalent to saying that for all x ∈ Ω,D2f(x) ≥ −M Id in the sense of symmetric matrices.

We will mainly use the notion of semiconvexity for the distance function to thefront. The following lemma will help us do so:

Lemma 2.5 ([29], Proposition 2.2.2). Let F 6= ∅ be a closed subset of RN . For all

convex subset Ω such that γ = infx∈Ω dF (x) > 0, the distance function to F , dF , issemiconcave on Ω, with constant 1/γ.

For the notion of subdifferential of a convex function, we refer to [65].

Definition 2.6. Let Ω be a convex open subset of RN and f : Ω → R be a convexfunction. The subdifferential of f at x ∈ Ω is the set

∂∗f(x) = p ∈ RN ; ∀y ∈ Ω, f(y) ≥ f(x) + 〈p, y − x〉.

If f is concave, the superdifferential of f at x ∈ Ω is the set

∂∗f(x) = p ∈ RN ; ∀y ∈ Ω, f(y) ≤ f(x) + 〈p, y − x〉.

These sets are never empty thanks to the separation theorem. If f is convex anddifferentiable at x, then ∂∗f(x) = Df(x). The following lemma is straightforwardbut will be useful:

Lemma 2.7 ([65]). Let fn, f be convex functions on a convex set Ω, satisfying:1. For any y ∈ Ω, lim sup fn(y) ≤ f(y),2. There exists a sequence (xn) converging to x such that lim fn(xn) = f(x),3. There exists a sequence (pn) such that pn ∈ ∂∗fn(xn) for all n and pn → p.

Then p ∈ ∂∗f(x).

51


2.3 BV functions and sets of finite perimeter

The results of this section are entirely taken from [38] unless explicitly statedotherwise. Let Ω be an open subset of RN .

2.3.1 Functions of bounded variations

Definition 2.8. A function f ∈ L1loc(Ω) is said to have locally bounded variations

in Ω if for any open subset U ⋐ Ω relatively compact in Ω,

sup

∫

U


< +∞.

We denote by BVloc(Ω) the set of functions of locally bounded variations in Ω. Wealso say that f ∈ L1(Ω) has bounded variations in Ω if the previous definition holdsfor U = Ω. We denote by BV (Ω) the set of functions of bounded variations in Ω.

The Riesz representation theorem then yields:

Theorem 2.9. Let f ∈ BVloc(Ω). Then there exists a Radon measure µ on Ω anda µ-measurable function σ : Ω → RN such that:

1. |σ(x)| = 1 µ-a.e.

2.∫Ωf(x) div φ(x) dx = −

∫Ω〈φ(x), σ(x)〉 dµ for all φ ∈ C1

c (Ω; RN).

The measure µ is called the variation measure of f , denoted by ‖Df‖, and we set[Df ] = σ ‖Df‖. We also have, for any open subset U ⋐ Ω relatively compact in Ω,

‖Df‖(U) = sup

∫

U


.

2.3.2 Sets of finite perimeter

Definition 2.10. An LN -measurable subset E ⊂ RN is said to have (locally) finiteperimeter in Ω if 1E has (locally) bounded variations in Ω. The variation measureof 1E in Ω is in this case denoted by ‖∂E‖, and the function −σ given by Theorem2.9 is denoted by νE.

Definition 2.11. Let E ⊂ RN be a set of locally finite perimeter in Ω. We say that

x ∈ Ω belongs to the reduced boundary of E, ∂∗E, if:

1. ‖∂E‖(B(x, r)) > 0 for all r > 0 such that B(x, r) ⊂ Ω,

2. 1‖∂E‖(B(x,r))

∫B(x,r)

νE d‖∂E‖ −→r→0

νE(x),

3. |νE(x)| = 1.

Then we have the following theorem:

Theorem 2.12 (Gauss-Green formula). Let E ⊂ RN be a set of locally finite perime-ter in Ω. Then for all φ ∈ C1

c (Ω; RN ),∫

E

div φ(x) dx =

∫

∂∗E

〈φ(x), νE(x)〉 dHN−1(x).

52

3. An integral formulation of the eikonal equation for subsolutions

Moreover we have control on the perimeter of the level sets of the distancefunction to a closed non-empty set, as a consequence of the following proposition(see [2, Lemma 2.4]):

Proposition 2.13. For all r0, r1 > 0 and R > 0, there exists M > 0 such that forall closed non-empty E ⊂ RN with diameter smaller than R, and for all r0 ≤ r ≤ r1,the set Er = x ∈ R

N ; dE(x) = r satisfies

HN−1(Er) ≤ M.

3 An integral formulation of the eikonal equation

for subsolutions

We first focus on the notion of subsolution. All the results can be adapted tothe case of supersolutions. The corresponding theorems and changes in proofs aregiven in Section 6.

Let c : RN × [0, T ] → R satisfy the assumptions of Theorem 2.2, and let the mappingK : [0, T ] → P(RN ) \ ∅ be such that

1. K(0) is compact, K(t) → K(0) as t → 0 and K(t) → K(T ) as t → T in theHausdorff distance.

2. The graph⋃

t∈[0,T ]t ×K(t) of K is closed in [0, T ] × RN .

3. u : (x, t) 7→ 1K(t)(x) is a viscosity subsolution of the eikonal equation

ut = c(x, t)|Du| in RN × (0, T ). (3.1)

Set, for all ε > 0,Kε(t) = x ∈ R

N ; dK(t)(x) < ε,

andcε(x, t) = max

|y−x|≤εc(y, t).

The aim of this section is the proof of the following result:

Theorem 3.1 (Integral formulation for subsolutions).For all t1 and t2 satisfying 0 ≤ t1 ≤ t2 ≤ T , for almost all ε > 0, and for all

φ ∈ C1(RN × [0, T ]; R+),

∫ t2

t1

∫

Kε(t)

φt(x, t) dxdt+

∫ t2

t1

∫

dK(t)=εcε(x, t)φ(x, t) dHN−1(x)dt

≥[∫

Kε(t)

φ(x, t) dx

]t2

t1

.

(3.2)

53


To this end, we set w(x, t) = −dK(t)(x) so that for all ε > 0,

Kε(t) = x ∈ RN ; w(x, t) > −ε.

Let us fix ε > 0 and θ : R → R satisfying the following conditions:

1. θ is non-decreasing and smooth,

2. θ(x) = 0 if x ≤ −ε, θ(x) = 1 if x ≥ 0.

Set wθ = θ w. We start by giving a semiconvexity property of wθ:

Lemma 3.2. For all (x, t) ∈ RN × (0, T ) such that x /∈ K(t), there exist M > 0,δ > 0 and r > 0 such that for all s ∈ (t − δ, t + δ), wθ(·, s) is semiconvex withconstant M on B(x, r).

Remark 3.3. In this lemma, the key point is the local uniformity with respect to sof the semiconvexity constant M .

Proof. Since x /∈ K(t), the lower semicontinuity of (x, t) 7→ dK(t)(x) given by Propo-sition 2.4 implies that there exist γ > 0, δ > 0 and r > 0 such that dK(s)(y) ≥ γ forevery s ∈ (t−δ, t+δ) and y ∈ B(x, r). But then thanks to Lemma 2.5, y 7→ dK(s)(y)is semiconcave on B(x, r) with constant 1/γ. The conclusion now follows from [29,Proposition 2.1.12] about the semiconvexity of the composite of a smooth functionwith a semiconvex function.

We are now ready to begin the proof of Theorem 3.1.

3.1 Equation satisfied by wθ in the viscosity sense

This subsection closely follows ideas of Soner [71] (see also [22, Theorem 3.1]).We provide the proof of the following proposition for the sake of completeness.

Proposition 3.4. wθ : (x, t) 7→ θ(−dK(t)(x)) is a subsolution of (wθ)t = cε(x, t)|Dwθ|in RN × (0, T ).

Proof. According to Proposition 2.4, w is usc on RN × [0, T ]. Since θ is continuousand non-decreasing, wθ is also usc.

Step 1. Let φ be of class C1 and such that wθ − φ attains a local maximum at(x0, t0) ∈ RN×(0, T ), which we can assume equal to 0, i.e. wθ(x0, t0) = φ(x0, t0). Lety0 be such that y0 ∈ K(t0) and w(x0, t0) = −|x0 − y0| (if x0 ∈ K(t0), then y0 = x0).Set ψ(z, t) = φ(z + x0 − y0, t). Let us show that u − ψ : (z, t) 7→ 1K(t)(z) − ψ(z, t)has a local maximum at (y0, t0), which will enable us to use the fact that u is asubsolution of (3.1). For z close to y0, and t close to t0,

−ψ(z, t) = −φ(z + x0 − y0, t) ≤ −wθ(z + x0 − y0, t)

because wθ−φ has a local maximum at (x0, t0) where it vanishes. Since by definitionwθ(z + x0 − y0, t) = θ(−dK(t)(z + x0 − y0)), for z ∈ K(t) we get

−ψ(z, t) ≤ −θ(−dK(t)(z + x0 − y0)) ≤ −θ(−|x0 − y0|)

54


by definition of dK(t), and because θ is non-decreasing. But

−θ(−|x0 − y0|) = −θ(w(x0, t0)) = −wθ(x0, t0) = −φ(x0, t0) = −ψ(y0, t0)

and 1K(t)(z) = 1K(t0)(y0) = 1, so u(z, t) − ψ(z, t) ≤ u(y0, t0) − ψ(y0, t0). Moreover,−ψ(z, t) → −ψ(y0, t0) as (z, t) → (y0, t0), thus in a neighborhood of (y0, t0),

−ψ(z, t) ≤ 1 − ψ(y0, t0). (3.3)

Therefore if z /∈ K(t), then 1K(t)(z) = 0, 1K(t0)(y0) = 1, and (3.3) means that,again, u(z, t) − ψ(z, t) ≤ u(y0, t0) − ψ(y0, t0), which is the desired result.

Step 2. Since u is a subsolution of (3.1), according to Step 1 we have the inequalityψt(y0, t0) ≤ c(y0, t0)|Dψ(y0, t0)|, that is to say

φt(x0, t0) ≤ c(y0, t0)|Dφ(x0, t0)|.This is where we see the interest of perturbing the equation with cε and of trun-cating with θ, because if dK(t0)(x0) = |x0 − y0| ≤ ε, then c(y0, t0) ≤ cε(x0, t0), andtherefore φt(x0, t0) ≤ cε(x0, t0)|Dφ(x0, t0)|. Besides, if on the contrary dK(t0)(x0) =−w(x0, t0) = |x0 − y0| > ε, since w is usc, it follows that w(x, t) < −ε in a neigh-borhood of (x0, t0), thus wθ(x, t) = 0 locally and the equation is still satisfied.

3.2 Variational equation satisfied by wθ

Proposition 3.5. For any test function φ ∈ C1c (R

N × (0, T ); R+),∫ T

0

∫

RN

wθ(x, t)φt(x, t) dxdt+

∫ T

0

∫

RN

cε(x, t)|Dwθ(x, t)|φ(x, t) dxdt ≥ 0. (3.4)

Proof. In order to prove this proposition, we carry out a technique of regularizationof wθ in time by sup-convolution: we define for σ > 0,

wσθ (x, t) = max

(y,s)∈RN×[0,T ]

wθ(y, s) −

1

σ[|x− y|2 + (t− s)2]

. (3.5)

This is justified: wθ is usc, hence (y, s) 7→ wθ(y, s)− 1σ[|x− y|2 +(t− s)2] is also usc.

In addition, wθ is bounded, so that

(y, s) 7→ wθ(y, s) −1

σ[|x− y|2 + (t− s)2]

is coercive. Consequently, the supremum is indeed a maximum. Moreover it is knownthat (3.5) defines a locally Lipschitz continuous function on RN×[0, T ] (while wθ wasonly Lipschitz continuous with respect to the space variable), converging pointwiseto wθ as σ converges to 0.

Step 1. Let us show, in the spirit of Step 1 of the previous proof, that wσθ is a

viscosity subsolution of (wσθ )t = [cε]h(σ) (x, t)|Dwσ

θ | in RN × (h(σ), T − h(σ)), whereh(σ) ∈ (0, 1), tends to 0 as σ → 0, and where we have set

[cε]α (x, t) = max|(y,s)−(x,t)|≤α

cε(y, s).

55


Let (x0, t0) ∈ RN × (0, T ). There exists (y0, s0) ∈ R

N × [0, T ] such that

wσθ (x0, t0) = wθ(y0, s0) −

1

σ[|x0 − y0|2 + (t0 − s0)

2].

We set u = (x0, t0), v = (y0, s0) for the sake of readability. Then

0 ≤ wσθ (u) = wθ(v) − 1

σ|u− v|2 ≤ 1 − 1

σ|u− v|2,

so |u − v| ≤ σ1/2 =: h(σ) and v ∈ RN × (0, T ) for u ∈ R

N × (h(σ), T − h(σ)).Therefore we can use the equation satisfied by wθ in RN × (0, T ) to deduce theequation satisfied by wσ

θ in RN × (h(σ), T − h(σ)), as in [51].

Step 2. Let φ ∈ C1c (R

N × (0, T ); R+), and U be an open and relatively compactsubset of RN × (0, T ) such that φ vanishes on U c. We can choose σ so small thatU ⊂ RN × (h(σ), T − h(σ)), and then wσ

θ is a viscosity subsolution of

(wσθ )t = [cε]h(σ) (x, t)|Dwσ

θ |in U thanks to Step 1. Since wσ

θ is locally Lipschitz continuous, it is differentiablealmost everywhere and the inequality (wσ

θ )t ≤ [cε]h(σ) (x, t)|Dwσθ | actually holds a.e.

in U (see [14]). Therefore,∫

RN×(0,T )

[(wσ

θ )t − [cε]h(σ) (x, t)|Dwσθ |]φ ≤ 0,

this integral being in fact taken on U . But wσθ is locally Lipschitz continuous and

therefore belongs to W 1,∞loc , and its a.e. time derivative coincides with its time

derivative in the sense of distributions. Thus∫

RN×(0,T )

wσθ φt + [cε]h(σ) (x, t)|Dwσ

θ |φ ≥ 0.

Now we want to pass to the limit in this expression as σ → 0 by applying thedominated convergence theorem.

Step 3. To do so we notice that wσθ φt and [cε]h(σ) φ are bounded on U uniformly

in σ since |wσθ | ≤ 1 and c is bounded. Moreover wσ

θ converges pointwise to wθ asσ → 0 as we recalled above, and [cε]h(σ) converges pointwise to cε. It only remainsto deal with the term |Dwσ

θ |.To this end, we fix a sequence (σn) converging to 0+. There exists a subset

U ⊂ U such that LN(U \ U) = 0 and Dwθ, Dwσnθ are defined on U for all n. Let us

show that (Dwσnθ ) converges a.e. to Dwθ on U as n goes to +∞:

Lemma 3.6. For almost all (x, t) ∈ U , Dwσn

θ (x, t) −→n→+∞

Dwθ(x, t).

Proof. First case: x /∈ K(t). Let us fix, thanks to Lemma 3.2, M > 0, δ > 0 andr > 0 such that for all s ∈ (t− δ, t+ δ), wθ(·, s) is semiconvex with constant M onB(x, r). For all n, choose (yn, sn) ∈ RN × (0, T ) realizing the supremum:

wσnθ (x, t) = wθ(yn, sn) − 1

σn[|x− yn|2 + (t− sn)2].

56


Recall that |(x, t) − (yn, sn)| ≤ h(σn) → 0, so that we can assume that for all n,

ψn : y 7→ wθ(y, sn) +M

2|y|2 and ψ : y 7→ wθ(y, t) +

M

2|y|2 (3.6)

are convex on B(x, r), and that yn ∈ B(x, r).

We know, thanks to Corollary 10.14 of [65], that Dwσnθ (x, t) = 2

σn(yn − x). Let

us compute Dwθ(yn, sn): for z close to yn,

wθ(z, sn) − 1

σn[|x− z|2 + (t− sn)2] ≤ wθ(yn, sn) − 1

σn[|x− yn|2 + (t− sn)2],

sowθ(z, sn) ≤ wθ(yn, sn) +

1

σn|x− z|2 − |x− yn|2.

But |x− z|2 − |x− yn|2 = |z − yn|2 + 2〈z − yn, yn − x〉, so we obtain

wθ(z, sn) ≤ wθ(yn, sn) +2

σn〈z − yn, yn − x〉 +

1

σn|z − yn|2.

Therefore 2σn

(yn −x) is a Fréchet superdifferential (see [29]) for wθ(·, sn) at yn. But,in addition, wθ(·, sn) is semiconvex on B(x, r) ∋ yn, whence differentiable at yn, andits gradient equals 2

σn(yn − x) (see [29, Proposition 3.1.5]). To sum up, for all n we

have Dwσnθ (x, t) = Dwθ(yn, sn) =: pn.

Now since sn → t and wθ is usc, for any z we have lim supψn(z) ≤ ψ(z), whereψn and ψ are defined by (3.6). Moreover, since wθ ≤ wσn

θ , and using the definitionof (yn, sn), we have

wθ(x, t) ≤ wθ(yn, sn) and lim supwθ(yn, sn) ≤ wθ(x, t),

which implies that wθ(yn, sn) → wθ(x, t) and

ψn(yn) = wθ(yn, sn) +M

2|yn|2 −→

n→+∞wθ(x, t) +

M

2|x|2 = ψ(x).

But pn = Dwσn

θ (x, t) = Dwθ(yn, sn), hence (pn) is bounded: for all s, wθ(s, ·) is‖θ′‖∞-Lipschitz continuous, so that ‖Dwθ‖∞ ≤ ‖θ′‖∞, and the same holds for Dwσn

θ

independently of n. We can extract a subsequence (nk) such that (pnk) converges

to some p ∈ RN . Then Dψnk(ynk

) = pnk+ Mynk

→ p + Mx. As a consequence,thanks to Lemma 2.7, we have p + Mx ∈ ∂∗ψ(x). But wθ is differentiable at(x, t) with respect to x, so ∂∗ψ(x) reduces to its gradient Dwθ(x, t) + Mx, whichshows that p + Mx = Dwθ(x, t) + Mx, and p = Dwθ(x, t). This holds for anyconverging subsequence of (pn), which proves that pn → Dwθ(x, t), that is to say,Dwσn

θ (x, t) → Dwθ(x, t) as n→ +∞.

Second case: x ∈ K(t). Since wθ ≤ wσnθ and wσn

θ ≤ maxwθ = 1, for all n wehave wσn

θ (x, t) = wθ(x, t) = 1. A result due to Stampacchia (see for instance [38]),asserts that Dwθ = Dwσn

θ = 0 almost everywhere on wθ = 1 since all wσnθ and

wθ are Lipschitz continuous with respect to the space variable. As a consequenceDwσn

θ (x, t) → Dwθ(x, t) as n→ +∞ for almost all (x, t) such that x ∈ K(t).

57


Finally, Dwσθ is bounded on U uniformly in σ: as we have seen above, we know

that ‖Dwσθ ‖∞ ≤ ‖θ′‖∞ for all σ > 0. The use of the dominated convergence theorem

is justified and proves Proposition 3.5.

3.3 The integral formulation

We are now ready to conclude the proof of Theorem 3.1. Fix 0 < α1 < α2 < ε.

Step 1. We first notice that an approximation argument shows that (3.4) also holdsfor the following function θ, although it is not of class C1:

1. θ is non-decreasing and continuous,

2. θ(x) = 0 if x ≤ −α2, θ(x) = 1 if x ≥ −α1,

3. θ is affine on [−α2,−α1].

Now we would like to transform the second term of (3.4) so as to get rid of |Dwθ|.To do so we interpret it as a jacobian thanks to the coarea formula (see [38]), whichyields, for all φ ∈ C1

c (RN × (0, T ); R+),

∫ T

0

∫

RN

wθ(x, t)φt(x, t) dxdt+

∫ T

0

∫ 1

0

∫

wθ(·,t)=τcε(x, t)φ(x, t) dHN−1(x)dτdt ≥ 0,

(3.7)where the integral from 0 to 1 represents the values taken by wθ(·, t).Step 2. Let us now transform each of the terms of (3.7) so as to get rid of wθ. Forthe first term, we simply notice, recalling that wθ(x, t) = θ(−dK(t)(x)), that

∫ T

0

∫

RN

wθ(x, t)φt(x, t) dxdt =

∫ T

0

∫

dK(t)<α2wθ(x, t)φt(x, t) dxdt

since θ(−σ) = 0 if σ ≥ α2. In the second term, let us make the change of variableτ = θ(−σ) for σ ∈ (α1, α2) and τ ∈ (0, 1). We observe that

wθ(x, t) = τ ⇔ θ(−dK(t)(x)) = θ(−σ) ⇔ dK(t)(x) = σ.

Therefore, we obtain for all φ ∈ C1c (R

N × (0, T ); R+) and for all α1, α2 satisfying0 < α1 < α2 < ε,

∫ T

0

∫

dK(t)<α2wθ(x, t)φt(x, t) dxdt

+1

α2 − α1

∫ α2

α1

∫ T

0

∫

dK(t)=σcε(x, t)φ(x, t) dHN−1(x)dtdσ ≥ 0,

(3.8)

where we have switched the integration order between t and σ, which is permittedsince ∫ T

0

∫ ε

0

∣∣∣∣∣

∫

dK(t)=σcε(x, t)φ(x, t) dHN−1(x)

∣∣∣∣∣ dσdt < +∞. (3.9)

58


Indeed we can estimate this expression by

‖c‖∞‖φ‖∞∫ T

0

∫ ε

0

HN−1(dK(t) = σ) dσdt.

Since |DdK(t)| = 1 a.e. on⋃

0<σ<εdK(t) = σ = Kε(t) \ K(t), the coarea formulashows that this integral is equal to

‖c‖∞‖φ‖∞∫ T

0

LN(Kε(t) \K(t)) dt.

But this last quantity is finite since the family K(t)t∈[0,T ] is uniformly bounded.

Step 3. We now use the freedom in the choice of α1 and α2 to deduce from (3.8) apointwise property of the integrand. To this end, we apply the Lebesgue-Besicovitchdifferentiation theorem (see for instance [38]) to the function

σ 7→∫ T

0

∫

dK(t)=σcε(x, t)φ(x, t) dHN−1(x)dt,

which lies in L1loc

(0, ε) for any φ ∈ C1c (RN × (0, T ); R+) thanks to (3.9). Fixing

σ ∈ (0, ε) and choosing α1 = σ − τ , α2 = σ + τ 2 in (3.8) with τ → 0, we deduce,since wθ(·, t) → 1Kσ(t) for this choice of α1 and α2, that for almost all σ ∈ (0, ε),

∫ T

0

∫

Kσ(t)

φt(x, t) dxdt+

∫ T

0

∫

dK(t)=σcε(x, t)φ(x, t) dHN−1(x)dt ≥ 0. (3.10)

Step 4. What we have done holds for a fixed φ ∈ C1c (RN ×(0, T ); R+) and ε > 0. We

now extend the result to all φ ∈ C1c (R

N × (0, T ); R+) and ε > 0, using the fact thatC1

c (RN × (0, T ); R+), equipped with the C1 norm, and R, are separable. This showsthat (3.10) holds for almost all σ > 0, all ε > σ and all φ ∈ C1

c (RN × (0, T ); R+).

Let us fix such a σ > 0. Sending ε to σ in (3.10) gives the integral formulation for testfunctions with compact support: for almost all ε > 0, for all φ ∈ C1

c (RN×(0, T ); R+),

∫ T

0

∫

Kε(t)

φt(x, t) dxdt+

∫ T

0

∫

dK(t)=εcε(x, t)φ(x, t) dHN−1(x)dt ≥ 0. (3.11)

Step 5. To conclude, it remains to generalize (3.11) to test functions without as-sumption on the support. Let 0 < t1 < t2 < T and α be a smooth non neg-ative function equal to 1 on [t1, t2] and having compact support in (0, T ). Letφ ∈ C1(RN × [0, T ]; R+) with compact support in the space variable. Applying(3.11) to αφ yields, for almost all ε > 0,

∫ T

0

∫

Kε(t)

α(t)φt(x, t) dxdt+

∫ T

0

∫

Kε(t)

αt(t)φ(x, t) dxdt

+

∫ T

0

∫

dK(t)=εcε(x, t)α(t)φ(x, t) dHN−1(x)dt ≥ 0,

59


which gives (3.2) when α converges to the indicator function of [t1, t2]:

∫ t2

t1

∫

Kε(t)

φt(x, t) dxdt−[∫

Kε(t)

φ(x, t) dx

]t2

t1

+

∫ t2

t1

∫

dK(t)=εcε(x, t)φ(x, t) dHN−1(x)dt ≥ 0.

This holds for almost all t1 and t2 in (0, T ) and therefore for all 0 < t1 ≤ t2 < Tsince 1Kε(t) → 1Kε(t0) in L1(RN) as t → t−0 thanks to Propositions 2.4 and 2.13(indeed, the latter guarantees that all sets of the form dK(t0) = ε have zero LN

measure). This being also true for t1 → 0+ and t2 → T− by assumption 1 beforeTheorem 3.1, we see that (3.2) also holds for t1 = 0 and t2 = T . Moreover, since theevolution is bounded, the time-dependent domain of integration Kε(t) is uniformlybounded. Thus if φ does not have compact support in the space variable either,after truncating φ in a C1 way off a large ball if necessary, we see that (3.2) holdsfor φ ∈ C1(RN × [0, T ]; R+). This concludes the proof of Theorem 3.1.

4 Conversely: from the integral formulation to the

notion of subsolution

In this section we are interested in the converse of Theorem 3.1:

Theorem 4.1. Let c : RN × [0, T ] → R satisfy the assumptions of Theorem 2.2 andK : [0, T ] → P(RN ) \ ∅ be such that

1. K is uniformly bounded on [0, T ].

2. The graph⋃


3. Inequality (3.2) holds for all 0 ≤ t1 ≤ t2 ≤ T , for almost all ε > 0 smallenough and for all φ ∈ C1(RN × [0, T ]; R+).

Then u : (x, t) 7→ 1K(t)(x) is a viscosity subsolution of ut = c(x, t)|Du| in RN×(0, T ).

During the proof of this theorem, we will need to use neighborhoods of a partic-ular form and the corresponding notion of open sets:

Definition 4.2. A set of the form B(x, r)× (t−h, t] with r > 0, h > 0 is called leftneighborhood of (x, t) ∈ RN × (0, T ). We say that U ⊂ RN × (0, T ) is left open if Ucontains a left neighborhood of each of its point.

We also define the corresponding notion of viscosity left subsolution, in whichthe test is restricted to left neighborhoods:

Definition 4.3. Let U ⊂ RN × (0, T ) be left open. We say that u is a viscosityleft subsolution of ut = c(x, t)|Du| in U if u is usc on RN × [0, T ] and if for all(x, t) ∈ U , for all test function φ of class C1 on RN × (0, T ) such that u− φ has amaximum on a left neighborhood of (x, t), we have φt(x, t) ≤ c(x, t)|Dφ(x, t)|.

60

4. Conversely: from the integral formulation to the notion of subsolution

It is easy to see that all classical results on viscosity solutions (stability, invariance,how a subsolution provides a set-theoretic subsolution) still hold for this notionof subsolution, under the only assumption in the stability theorem that the upperrelaxed semi-limit lim sup∗(un) of (un) (see [14]) satisfies

lim sup∗(un)(x, t) = lim supun(xn, tn); tn → t, tn ≤ t, xn → x,

i.e. we ask that the lim sup be achieved through lower times. We are now ready tobegin the proof of Theorem 4.1.

Proof. The fact that⋃

t∈[0,T ]t × K(t) is closed in [0, T ] × RN ensures that themapping (x, t) 7→ 1K(t)(x) is usc. For the rest of the proof, let us fix ε0 such that(3.2) holds for almost all ε ∈ (0, ε0).

Step 1. Let us first prove that for all (x0, t0) ∈ RN × (0, T ) such that x0 ∈ Kε0(t0),t 7→ dK(t)(x0) is left continuous at t0. If this were not true, there would existt0 ∈ (0, T ), x0 ∈ Kε0(t0), η > 0 and a sequence (tn) converging to t−0 such that forall n ≥ 1,

dK(t0)(x0) + η < dK(tn)(x0). (4.1)

Let us choose α ∈ (0, η/2) and φ ∈ C1c (B(x0, α); R+) with φ(x0) > 0. Assumption

3 implies that (3.2) holds with ε = dK(t0)(x0) + δ for almost all δ ∈ (0, η/2) smallenough. Let us apply it to the time-independent function (x, t) 7→ φ(x) between tnand t0 for such a δ:

∫ t0

tn

∫

dK(t)=εcε(x, t)φ(x) dHN−1(x)dt ≥

[∫

Kε(t)

φ(x) dx

]t0

tn

.

But [∫

Kε(t)

φ(x) dx

]t0

tn

=

∫

Kε(t0)

φ(x) dx,

because (4.1) implies that Kε(tn) ∩B(x0, α) = ∅ for all n ≥ 1. Moreover

∫ t0

tn

∫

dK(t)=εcε(x, t)φ(x) dHN−1(x)dt ≤ (t0 − tn)M ‖c‖∞ ‖φ‖∞,

where M denotes the bound on HN−1(dK(t) = ε) given by Proposition 2.13. Send-ing n to +∞, we deduce that

∫

Kε(t0)

φ(x) dx ≤ 0,

which in view of the choice of φ implies that x0 /∈ Kε(t0), i.e.

dK(t0)(x0) ≥ ε = dK(t0)(x0) + δ.

This is absurd and proves the claim.

61


Step 2. A straightforward consequence of (3.11) is that, under assumption 3, wealso have, for almost all ε ∈ (0, ε0), σ ∈ (0, ε), and for all φ ∈ C1

c (RN × (0, T ); R+),

∫ T

0

∫

Kσ(t)

φt(x, t) dxdt+

∫ T

0

∫

dK(t)=σcε(x, t)φ(x, t) dHN−1(x)dt ≥ 0, (4.2)

since φ ≥ 0 and cσ ≤ cε. Let us integrate inequality (4.2) for σ between 0 and somefixed ε ∈ (0, ε0) and switch the order of integration between t and σ:

∫ T

0

∫ ε

0

∫

Kσ(t)

φt(x, t) dxdσdt+

∫ T

0

∫ ε

0

∫

dK(t)=σcε(x, t)φ(x, t) dHN−1(x)dσdt ≥ 0.

(4.3)This inversion is justified because of (3.9), and the coarea formula shows that

∫ ε

0

∫

dK(t)=σcε(x, t)φ(x, t) dHN−1(x)dσ =

∫

Kε(t)\K(t)

cε(x, t)φ(x, t) dx.

Now switching the order between x and σ in the first term of (4.3), we get∫ ε

0

∫

Kσ(t)

φt(x, t) dxdσ =

∫

Kε(t)

(ε− dK(t)(x))φt(x, t) dx, (4.4)

and we deduce from the last three equations that

∫ T

0

∫

Kε(t)

(ε− dK(t)(x))φt(x, t) dxdt+

∫ T

0

∫

Kε(t)\K(t)

cε(x, t)φ(x, t) dxdt ≥ 0. (4.5)

Separating the test function into (x, t) 7→ θn(t)φ(x), with θn and φ of class C1, andsending θn to θ = 1[t1,t2] with 0 < t1 < t2 < T , we deduce from (4.5) that

∫

Kε(t2)

(ε− dK(t2)(x))φ(x) dx

≤∫

Kε(t1)

(ε− dK(t1)(x))φ(x) dx+

∫ t2

t1

∫

Kε(t)\K(t)

cε(x, t)φ(x) dxdt.

(4.6)

This holds for almost all t1 and t2, and therefore for all 0 < t1 ≤ t2 < T thanks toStep 1 and the fact that all sets of the form dK(t0) = ε have zero LN measure.Seeing (4.6), we could be tempted to try to prove that w : (x, t) 7→ −dK(t)(x) is asubsolution of wt = cε(x, t)|Dw| in

Aε = (x, t) ∈ RN × (0, T ) ; 0 < dK(t)(x) < ε,

so that if (x, t) ∈ Aε, x ∈ Kε(t) \K(t). Unfortunately, this last assertion does notmake sense since Aε is not open. Indeed K(t) may “shrink” suddenly as t increases.However, Step 1 simplies that Aε is left open. Indeed if (x, t) ∈ Aε, then s 7→ dK(s)(x)is left continuous at t with 0 < dK(t)(x) < ε.

62

4. Conversely: from the integral formulation to the notion of subsolution

Let us now go back to (4.6), and show that w : (x, t) 7→ −dK(t)(x) is a viscosityleft subsolution of wt = cε(x, t)|Dw| in Aε. Take (x, t) ∈ Aε and ψ of class C1 onRN × (0, T ) such that w − ψ has a maximum equal to 0 on a left neighborhood of(x, t). Then for certain r > 0 and h > 0, and for all (x, t) ∈ B(x, r)× [ t−h, t ] ⊂ Aε,

−dK(t)(x) ≤ ψ(x, t), (4.7)

the inequality being an equality at (x, t). Taking t1 = t − h, t2 = t in (4.6), andusing the freedom in the choice of φ to send φ to the Dirac mass at x, yields

ε− dK(t)(x) ≤ ε− dK(t−h)(x) +

∫ t

t−h

cε(x, t) dt.

Then we can use (4.7) to obtain

ψ(x, t) ≤ ψ(x, t− h) +

∫ t

t−h

cε(x, t) dt.

Dividing by h > 0 and sending h to 0+ gives −ψt(x, t) + cε(x, t) ≥ 0 which meansthat w is a left subsolution of wt = cε(x, t)|Dw| in Aε, since |Dψ(x, t)| = 1. Indeed−dK(t) is locally semiconvex around x thanks to Lemma 2.5, and smaller than ψ(·, t),which is smooth, with equality at x. Therefore −dK(t) is differentiable at x with−DdK(t)(x) = Dψ(x, t), but in Aε, |DdK(t)(x)| = 1 whenever this gradient exists.

As a consequence of the stability theorem, 1w≥−ε/2 is a left subsolution ofwt = cε(x, t)|Dw| in Aǫ, and therefore also in RN × (0, T ). Indeed, if x /∈ Kε(t),w(x, t) < −3

4ε, which remains true in a neighborhood of (x, t) since w is usc. Thus

1w≥−ε/2(y, s) = 0 in a neighborhood of (x, t). Moreover if x ∈ K(t), w(x, t) > −14ε,

which this time does not necessarily hold on a neighborhood, but remains true on aleft neighborhood of (x, t) since w is left continuous. Therefore 1w≥−ε/2(y, s) = 1on a left neighborhood of (x, t). In both cases, if 1w≥−ε/2 − ψ has a maximum ona left neighborhood of (x, t) with ψ of class C1, then ψt(x, t) ≤ 0 = |Dψ(x, t)|, sothe equation is satisfied in the viscosity sense.

We have just proved that 1w≥−ε/2 is a left subsolution of wt = cε(x, t)|Dw| inRN × (0, T ). But RN × (0, T ) is open in the usual sense, so 1w≥−ε/2 actually isa viscosity subsolution of wt = cε(x, t)|Dw| in RN × (0, T ) without restriction onthe neighborhoods, since a local maximum is in particular a maximum on some leftneighborhood.

Now since w(x, t) ≥ −ε/2 ⇔ dK(t)(x) ≤ ε/2, the stability theorem shows, asε tends to 0, that u : (x, t) 7→ 1K(t)(x) is a subsolution of ut = c(x, t)|Du| inRN × (0, T ), which is the desired conclusion. Indeed cε converges uniformly to csince c is Lipschitz continuous with respect to x, and if εn → 0+ is fixed, then

1K(t)(x) = lim sup1dK(tn)≤εn/2(xn); tn → t, xn → x, εn → 0.

In order to see that, let us fix a sequence (xn, tn) converging to (x, t). We noticethat if x ∈ K(t), 1K(t)(x) = 1 ≥ sup 1dK(tn)≤εn/2(xn). Moreover if x /∈ K(t),

63


(x, t) does not belong to the graph of K which is closed, and thus is located at apositive distance from this graph, which implies that for n large enough, we have1K(t)(x) = 0 = 1dK(tn)≤εn/2(xn). Therefore

1K(t)(x) ≥ lim sup1dK(tn)≤εn/2(xn); tn → t, xn → x, εn → 0.

To conclude it suffices to construct a sequence (xn, tn) converging to (x, t) such that1dK(tn)≤εn/2(xn) → 1K(t)(x). Take xn = x and tn = t. If x /∈ K(t), we havejust seen that 1K(t)(x) = 0 = 1dK(t)≤εn/2(x) for n large enough. If x ∈ K(t), thenx ∈ Kεn/2(t) so that 1dK(t)≤εn/2(x) = 1 = 1K(t)(x).

5 Regularity of the front

In this section we use the integral formulation to derive estimates related to theregularity of K(t), and more precisely its perimeter, by studying the limit of (3.2)as ε tends to 0. To this end we make the following assumptions in addition of thoseof Theorem 2.2:

(A1) c is of class C1, Dc is locally Lipschitz continuous with respect to the spacevariable.

(A2) Dc(x, t) 6= 0 if c(x, t) = 0.

Theorem 5.1. Let c : RN × [0, T ] → R satisfy the assumptions of Theorem 2.2 and(A1), (A2). Let K : [0, T ] → P(RN) \ ∅ be such that:

1. K(0) is compact, K(t) → K(0) as t → 0 and K(t) → K(T ) as t → T in theHausdorff distance.

2. The graph⋃


3. u : (x, t) 7→ 1K(t)(x) is a subsolution of the eikonal equation (3.1).

Then the following statements hold:(i) For a.e. t ∈ [0, T ], c(·, t)1K(t) has bounded variations in c(·, t) < 0.(ii) For a.e. t ∈ [0, T ], K(t) has locally finite perimeter in c(·, t) < 0.(iii) Denoting (·)− the negative part of a quantity (x− = max(−x, 0)), we have:

∫ T

0

∫

∂∗K(t)

c−(x, t) dHN−1(x)dt < +∞,

an upper bound for this integral being given by

−[∫

K(t)

1c<0(x, t) dx

]T

0

+

∫ T

0

∫

K(t)∩c(·,t)=0

|ct(x, t)||Dc(x, t)| 1ct<0(x, t) dHN−1(x)dt.

Proof of Theorem 5.1 (i): we split this rather long proof into several lemmas.In what follows, BR = B(0, R) will denote a large ball which contains Kε(t) for allt ∈ [0, T ] and ε ≤ 1, and Kε(t) stands for Kε(t).

64

5. Regularity of the front

Lemma 5.2. For all θ > 0 small enough, for almost all 0 < ε ≤ ε0 ≤ 1,

∫ T

0

∫

Kε(t)∩c(·,t)=−θ

|ct(x, t)||Dc(x, t)| 1ct<0(x, t) dHN−1(x)dt (5.1)

+

∫ T

0

∫

dK(t)=εcε0(x, t)1c<−θ(x, t) dHN−1(x)dt ≥

[∫

Kε(t)

1c<−θ(x, t) dx

]T

0

.

Proof. Let us fix θ > 0 and 0 < ε ≤ ε0 ≤ 1 such that (3.2) holds with this ε. For allη ∈ (0, θ), let Tη be a smooth non-increasing function equal to 1 in (−∞,−θ − η]and 0 in [−θ,+∞). Then (3.2) can be applied to φη = Tη c with t1 = 0 andt2 = T . Now an approximation argument shows that (3.2) also holds for Tη definedas follows:

1. Tη is non-increasing and continuous,

2. Tη(s) = 1 if s ≤ −θ − η, Tη(s) = 0 if s ≥ −θ,3. Tη is affine on [−θ − η,−θ],

and φη = Tη c. Since (φη)t = ct (T′η c) = −1

ηct 1−θ−η<c<−θ, we get

∫ T

0

∫

Kε(t)∩−θ−η<c(·,t)<−θ

(−1

η

)ct(x, t) dxdt

+

∫ T

0

∫

dK(t)=εcε(x, t)φη(x, t) dHN−1(x)dt ≥

[∫

Kε(t)

φη(x, t) dx

]T

0

.

(5.2)

We want to send η to 0 in (5.2). Thanks to the coarea formula,

∫ T

0

∫


(−1

η

)ct(x, t) dxdt

≤∫ T

0

∫


1

η|ct(x, t)| 1ct<0(x, t) dxdt

=1

η

∫ T

0

∫ −θ

−θ−η

∫

Kε(t)∩c(·,t)=σ

|ct(x, t)||Dc(x, t)| 1ct<0(x, t) dHN−1(x)dσdt.

(5.3)

The coarea formula can be applied since c is Lipschitz continuous with respect tothe space variable and of class C1, so that assumption (A2) implies that if θ is smallenough and η ∈ (0, θ), then Dc is bounded away from 0 on the set

−θ − η < c < −θ ∩ BR × [0, T ].

As a standard consequence of assumptions (A1) and (A2), there exists σ0 > 0 suchthat the map

σ 7→∫ T

0

∫

Kε(t)∩c(·,t)=σ

|ct(x, t)||Dc(x, t)| 1ct<0(x, t) dHN−1(x)dt

65


is usc on (−σ0, σ0) for all 0 ≤ ε ≤ 1 (with the convention that K0(t) = K(t)). Inparticular for θ small enough, as η tends to 0, the last term of (5.3) satisfies

lim supη→0

1

η

∫ T

0

∫ −θ

−θ−η

∫

Kε(t)∩c(·,t)=σ

|ct(x, t)||Dc(x, t)| 1ct<0(x, t) dHN−1(x)dσdt

≤∫ T

0

∫

Kε(t)∩c(·,t)=−θ


(5.4)

Moreover, for ε ≤ ε0,∫ T

0

∫

dK(t)=εcε(x, t)φη(x, t) dHN−1(x)dt

≤∫ T

0

∫

dK(t)=εcε0(x, t)φη(x, t) dHN−1(x)dt

−→η→0

∫ T

0

∫

dK(t)=εcε0(x, t) 1c<−θ(x, t) dHN−1(x)dt,

(5.5)

and [∫

Kε(t)

φη(x, t) dx

]T

0

−→η→0

[∫

Kε(t)

1c<−θ(x, t) dx

]T

0

. (5.6)

Combining (5.2) to (5.6) then gives (5.1).

Intermediate step. Let Ωt = x ∈ RN ; c(x, t) < −θ for θ chosen small enough sothat (5.1) holds. Since c is of class C1, for ε0 ≤ 1 small enough depending on θ,cε0 < 0 on

⋃t∈[0,T ](BR ∩Ωt)×t, so cε0 = −(cε0)− on this set. Since in addition cε0

is Lipschitz continuous with respect to the space variable because of its definition(with ‖Dcε0‖∞ ≤ ‖Dc‖∞), we see by a regularization argument that there exists csatisfying the following assumptions:

1. c is of class C1 on RN × [0, T ],

2. c is Lipschitz continuous with respect to the space variable,

with ‖Dc‖∞ ≤ ‖Dcε0‖∞,3. 0 ≤ c ≤ (cε0)− in

⋃

t∈[0,T ]

(BR ∩ Ωt) × t.(5.7)

Then (5.1) gives, for almost all 0 < ε ≤ ε0 and all c satisfying (5.7),

∫ T

0

∫

dK(t)=εc(x, t) 1c<−θ(x, t) dHN−1(x)dt

≤∫ T

0

∫

dK(t)=ε(cε0)−(x, t) 1c<−θ(x, t) dHN−1(x)dt (5.8)

≤ −[∫

Kε(t)

1c<−θ(x, t) dx

]T

0

+

∫ T

0

∫

Kε(t))∩c(·,t)=−θ


66


Let us fix c satisfying (5.7). Now we want, thanks to the control given by (5.8), toestimate the total variation of c(·, t) 1Kε(t), and then send ε to 0 to get an estimateon c(·, t) 1K(t). To this end, we introduce aε(x, t) = c(x, t) 1Kε(t)(x). Proposition3.2 of [8] on the product of a BV function by a regular function shows that for allt ∈ [0, T ] and all ε > 0, aε(·, t) ∈ BV (RN), and if [Daε(·, t)] denotes the vector-valued variation measure of aε(·, t) as defined in Theorem 2.9, we have

[Daε(·, t)] = −c(·, t) νKε(t) HN−1⌊∂∗Kε(t) + 1Kε(t) Dc(·, t)LN . (5.9)

Now let us introduce two notations:

1. For a fixed t ∈ [0, T ] and θ ≥ 0, Xθt denotes the set of vector-valued functions

φ ∈ C1(BR; RN) vanishing on BR ∩ Ωct , with ‖φ‖∞ ≤ 1.

2. Xθ is the set of all functions φ such that φ(·, t) ∈ Xθt for all t ∈ [0, T ], with φ

and Dφ measurable on RN × [0, T ], and ‖Dφ‖∞ < +∞.

Lemma 5.3. For all θ > 0 small enough,

supφ∈Xθ

∫ T

0

∫

K(t)

c−(x, t)1c<−θ(x, t) divx φ(x, t) dxdt

≤M(θ), (5.10)

where divx stands for the divergence with respect to the space variable, and where

M(θ) = LN(K(0)) + T LN(BR) ‖Dc‖∞

+

∫ T

0

∫

K(t)∩c(·,t)=−θ


Proof. Fix θ > 0 small enough and choose ε0 ≤ 1 so that (5.8) holds for almostall ε ≤ ε0 and all c satisfying (5.7). Let φ ∈ Xθ. Since aε(·, t) vanishes off BR forε ≤ ε0, we have thanks to (5.9), after extending φ(·, t) to RN in a C1 way so as toobtain a function with compact support in the space variable:

∫ T

0

∫

RN

aε(x, t) divx φ(x, t) dxdt

=

∫ T

0

∫

∂∗Kε(t)

c(x, t) 1c<−θ(x, t) 〈φ(x, t), νKε(t)(x)〉 dHN−1(x)dt

−∫ T

0

∫

Kε(t)

1c<−θ(x, t) 〈Dc(x, t), φ(x, t)〉 dxdt.

(5.11)

The integration in time is justified since each of the expressions under the first andthird time integral signs is integrable on (0, T ) by Fubini’s theorem. For φ ∈ Xθ,

∫ T

0

∫

∂∗Kε(t)


≤∫ T

0

∫


(5.12)

67


since ∂∗Kε(t) ⊂ ∂Kε(t) = dK(t) = ε, and 1c<−θ(x, t) c(x, t) ≥ 0 thanks to (5.7),point 3. But (5.8) precisely gives an upper bound for this term independently of εfor almost all 0 < ε ≤ ε0:

∫ T

0

∫


≤LN(Kε0(0)) +

∫ T

0

∫

Kε0(t)∩c(·,t)=−θ


(5.13)

In addition, for φ ∈ Xθ and ε ≤ ε0,∣∣∣∣∫ T

0

∫

Kε(t)

1c<−θ(x, t) 〈Dc(x, t), φ(x, t)〉 dxdt∣∣∣∣

≤T LN(BR) ‖Dc‖∞ ≤ T LN(BR) ‖Dcε0‖∞ ≤ T LN(BR) ‖Dc‖∞(5.14)

because of (5.7), point 2, and by definition of cε0.

Let us introduce a variation measure on (x, t) ∈ RN × [0, T ]; c(x, t) < −θ withrespect to the space variable by the following formula:

‖Dxaε‖ = supφ∈Xθ

∫ T

0

∫

RN

aε(x, t) divx φ(x, t) dxdt

.

We deduce from equations (5.11) to (5.14) that ε 7→ ‖Dxaε‖ is bounded on (0, ε0].Moreover (aε) converges to a : (x, t) 7→ c(x, t) 1K(t)(x) in L1 as ε tends to 0. But itis straightforward to see that ‖Dx(·)‖ is lsc for the topology of L1(RN × (0, T ); R)(see [38, Theorem 1 p. 172]). This implies that a has finite ‖Dx(·)‖ variation, with

‖Dxa‖ ≤ lim infε→0

‖Dxaε‖≤ LN(K(0)) + T LN(BR) ‖Dc‖∞

+

∫ T

0

∫

K(t)∩c(·,t)=−θ

|ct(x, t)||Dc(x, t)| 1ct<0(x, t) dHN−1(x)dt = M(θ).

The integral term in the last inequality comes when sending ε0 to 0 in (5.13) bydominated convergence. Indeed Kε0(t) → K(t) everywhere and the dominationcomes from the fact that Dc is bounded away from 0 on c = −θ ∩ BR × [0, T ].The inequality ‖Dxa‖ ≤M(θ) amounts to saying that for all c satisfying (5.7),

supφ∈Xθ

∫ T

0

∫

K(t)

c(x, t) 1c<−θ(x, t) divx φ(x, t) dxdt

≤M(θ).

Now a convolution argument shows that we can make c converge uniformly to (cε0)−on BR × [0, T ] with c satisfying (5.7). Therefore going to the limit in the previousinequality by dominated convergence, and then finally sending ε0 to 0 by the sameargument, we get the result.

68


We are finally ready to send θ to 0. Set X = X0 and for all t ∈ [0, T ], Xt = X0t ,

where X0 and X0t are defined before Lemma 5.3. In particular any φ ∈ X vanishes

outside c < 0.

Lemma 5.4.

supφ∈X

∫ T

0

∫

K(t)

c(x, t) divx φ(x, t) dxdt

≤M(0). (5.15)

Proof. Let φ ∈ X. Consider the truncature function T θ, a continuous and piecewiseaffine function, such that T θ(s) = 1 for s ≤ −2θ, and T θ(s) = 0 for s ≥ −θ. Weregularize T θ to get a sequence (T θ

n) of functions converging uniformly to T θ, whosederivatives converge pointwise to (T θ)′, and such that there exists a constant C > 0satisfying ‖(T θ

n)′‖∞ ≤ C/θ for all n. Then (5.10) can be applied to (T θn c)φ. But

divx (T θn c)φ = ((T θ

n)′ c) 〈φ,Dc〉+ (T θn c) divx φ.

As n goes to infinity, we deduce from the dominated convergence theorem that

∫ T

0

∫

K(t)

c−(x, t)

(−1

θ

)1−2θ<c<−θ(x, t) 〈φ(x, t), Dc(x, t)〉 dxdt

+

∫ T

0

∫

K(t)

c−(x, t) (T θ c) divx φ(x, t) dxdt ≤M(θ).

As θ tends to 0, the first term converges to 0: the domination comes from the factthat |c−(x, t) (−1

θ) 1−2θ<c<−θ(x, t)| ≤ 1

θ(2θ) = 2 and the domain of integration is

bounded. The second term converges to

∫ T

0

∫

K(t)

c−(x, t) divx φ(x, t) dxdt.

Moreover, recall that θ 7→ M(θ) is usc on (−σ0, σ0) for σ0 small enough, so thatlim supθ→0 M(θ) ≤M(0). Given that c− = −c on c < 0, we get the desired resultin the limit as θ → 0.

The following lemma is a consequence of (5.15) and will be proved at the end ofthis section for the sake of readability:

Lemma 5.5. For almost all t ∈ [0, T ],

supφ∈Xt

∫

K(t)

c(x, t) div φ(x) dx

< +∞.

We deduce from this lemma and the definition of Xt that for almost all t ∈ [0, T ],c(·, t) 1K(t) has bounded variations in c(·, t) < 0, and assertion (i) of Theorem 5.1is finally proved.

69


Proof of Theorem 5.1 (ii).

Let us fix a t such that c(·, t) 1K(t) has bounded variations in c(·, t) < 0. For allη > 0, 1/c(·, t) is of class C1 on x ∈ RN ; c(x, t) < −η, and Lipschitz continuous.It follows from Proposition 3.2 of [8] that 1K(t) has bounded variations, whence K(t)has finite perimeter, in x ∈ RN ; c(x, t) < −η for all η > 0. This proves assertion(ii) of Theorem 5.1: K(t) has locally finite perimeter in c(·, t) < 0 for almost allt ∈ [0, T ].

Proof of Theorem 5.1 (iii).

Step 1. As in the intermediate step before Lemma 5.3, let us fix θ > 0 small enoughso that (5.1) holds, and take ε0 ≤ 1 small enough depending on θ and c satisfying(5.7). Now (5.9) can be applied to a : (x, t) 7→ c(x, t)1K(t) and shows that for almostall t ∈ [0, T ],

[Da(·, t)] = −c(·, t) νK(t) HN−1⌊∂∗K(t) + 1K(t)Dc(·, t)LN in c(·, t) < 0,

which amounts to saying that for almost all t ∈ [0, T ], for all φ of class C1 on RN

with compact support in c(·, t) < 0,∫

K(t)


=

∫

∂∗K(t)

c(x, t) 〈φ(x), νK(t)(x)〉 dHN−1(x) −∫

K(t)

〈Dc(x, t), φ(x)〉 dx.(5.16)

Now let φ be of class C1 on BR and vanish off c(·, t) < −θ for some θ > 0. We canassume without changing any of the integrals in (5.16) that φ has compact supportin c(·, t) < 0, so that (5.16) holds for φ.

Step 2. Fix φ ∈ Xθ. Thanks to (5.16) we have

∫ T

0

∫

K(t)

c(x, t) 1c<−θ(x, t) divx φ(x, t) dxdt

=

∫ T

0

∫

∂∗K(t)

c(x, t) 1c<−θ(x, t) 〈φ(x, t), νK(t)(x)〉 dHN−1(x)dt

−∫ T

0

∫

K(t)

1c<−θ(x, t) 〈Dc(x, t), φ(x, t)〉 dxdt.

(5.17)

Let us go back to (5.11): by dominated convergence, the first and third term of thisequality respectively converge as ε → 0 to the first and third term of (5.17), whichshows that

∫ T

0

∫

∂∗Kε(t)


−→ε→0

∫ T

0

∫

∂∗K(t)

c(x, t) 1c<−θ(x, t) 〈φ(x, t), νK(t)(x)〉 dHN−1(x)dt.

70


Therefore, sending ε to 0, c to (cε0)− and ε0 to 0 in (5.8) now shows, since we have∂∗Kε(t) ⊂ ∂Kε(t) = dK(t) = ε, that∫ T

0

∫

∂∗K(t)

c−(x, t) 1c<−θ(x, t) 〈φ(x, t), νK(t)(x)〉 dHN−1(x)dt (5.18)

≤ −[∫

K(t)

1c<−θ(x, t) dx

]T

0

+

∫ T

0

∫

K(t)∩c(·,t)=−θ


We will need two additional notations:

1. For a fixed t ∈ [0, T ] and θ ≥ 0, Xθt denotes the set of vector-valued functions

φ ∈ C0(BR; RN) vanishing on BR ∩ Ωct , with ‖φ‖∞ ≤ 1.

2. Xθ is the set of all functions φ such that φ(·, t) ∈ Xθt for all t ∈ [0, T ], with φ

measurable on RN × [0, T ].

An approximation argument shows that (5.18) holds for φ ∈ Xθ.

Step 3. We give a lemma in the spirit of Lemma 5.5, the proof of which will also begiven at the end of this section:

Lemma 5.6. For all θ > 0:

supφ∈Xθ

∫ T

0

∫

∂∗K(t)

c−(x, t)1c<−θ(x, t) 〈φ(x, t), νK(t)(x)〉 dHN−1(x)dt

=

∫ T

0

supφ∈Xθ

t

∫

∂∗K(t)

c−(x, t)1c<−θ(x, t) 〈φ(x), νK(t)(x)〉 dHN−1(x)

dt.

(5.19)

We keep in mind that (5.18) gives an upper bound for the left-hand side of thisequality. Now we show that for all θ > 0 and t ∈ [0, T ] such that K(t) has finiteperimeter in x ∈ RN ; c(x, t) < −θ (which is true for almost all t thanks to theproof of Theorem 5.1 (ii)), the supremum actually equals

∫

∂∗K(t)

c−(x, t) 1c<−θ(x, t) dHN−1(x). (5.20)

Let such θ > 0 and t ∈ [0, T ] be fixed. Let C ⊂ c(·, t) < −θ be compact. SinceK(t) has finite perimeter in x ∈ RN ; c(x, t) < −θ, C ∩ ∂∗K(t) is a set of finiteHN−1 measure. Therefore the integral

∫

C∩∂∗K(t)

c−(x, t) dHN−1

is finite and according to Lusin’s theorem (see [38] for instance), we know that forall n ≥ 1, there exists a compact set Cn ⊂ C ∩ ∂∗K(t) such that νK(t) is continuouson Cn and

HN−1([C ∩ ∂∗K(t)] \ Cn) < 1/n. (5.21)

We fix n ≥ 1. For all η > 0 small enough so that C + B(0, η) ⊂ c(·, t) < −θ, weconsider φη such that φη ∈ C0

c (Cn + B(0, η)), ‖φη‖ ≤ 1 and φη coincides with νK(t)

71


on Cn. The existence of φη is guaranteed by [38, Theorem 1 p. 13]. Then φη ∈ Xθt

and∫

∂∗K(t)

c−(x, t) 〈φη(x), νK(t)(x)〉 dHN−1(x)

=

∫

Cn

c−(x, t) dHN−1(x) +

∫

∂∗K(t)

1[Cn+B(0,η)]\Cn(x) c−(x, t) 〈φη(x), νK(t)(x)〉 dHN−1(x).

As η goes to 0,∣∣∣∣∫

∂∗K(t)

1[Cn+B(0,η)]\Cn(x) c−(x, t) 〈φη(x), νK(t)(x)〉 dHN−1(x)

∣∣∣∣

≤ ‖c‖∞∫

∂∗K(t)

1[Cn+B(0,η)]\Cn(x) dHN−1(x) → 0

thanks to the dominated convergence theorem. Therefore∣∣∣∣∫

∂∗K(t)

c−(x, t) 〈φη(x), νK(t)(x)〉 dHN−1(x) −∫

C∩∂∗K(t)

c−(x, t) dHN−1(x)

∣∣∣∣

−→η→0

∫

[C∩∂∗K(t)]\Cn

c−(x, t) dHN−1(x) ≤ ‖c‖∞1

n

thanks to (5.21). We have constructed functions φη ∈ Xθt such that the integral

∫

∂∗K(t)

c−(x, t) 〈φη(x), νK(t)(x)〉 dHN−1(x)

is arbitrarily close to the integral∫

C∩∂∗K(t)c−(x, t) dHN−1(x), which in turn, by

takingC = Cp = B(0, p) ∩ x; dc(·,t)≥−θ(x) ≥ 1/p

as p→ +∞, can be made arbitrarily close to∫

∂∗K(t)

c−(x, t) 1c<−θ(x, t)dHN−1(x),

using the Beppo-Levi monotone convergence theorem. This proves (5.20).

Step 4. Combining the results of Steps 2 and 3, we have, for all θ > 0,∫ T

0

∫

∂∗K(t)

c−(x, t) 1c<−θ(x, t) dHN−1(x)dt

≤ −[∫

K(t)

1c<−θ(x, t) dx

]T

0

+

∫ T

0

∫

K(t)∩c(·,t)=−θ


If we let θ to 0 in this estimation, since the right-hand side is upper semicontinuous,the monotone convergence theorem shows that

∫ T

0

∫

∂∗K(t)

c−(x, t) dHN−1(x)dt

72


is finite and

∫ T

0

∫

∂∗K(t)

c−(x, t) dHN−1(x)dt

≤ −[∫

K(t)

1c<0(x, t) dx

]T

0

+

∫ T

0

∫

K(t)∩c(·,t)=0

|ct(x, t)||Dc(x, t)| 1ct<0(x, t) dHN−1(x)dt,

which finally proves the third assertion of Theorem 5.1.

We end with the proofs of the two lemmas we had postponed.

Proof of Lemma 5.5. If the lemma were not true, there would exist A ⊂ [0, T ]with positive measure such that for all t ∈ A,

supφ∈Xt

∫

K(t)


= +∞.

Let ε > 0 be fixed. For all M > 0, set

EM = t ∈ A; ∃φ ∈ Xt with ‖Dφ‖∞ ≤M,

∫

K(t)

c(x, t) div φ(x) dx ≥ 1/ε,

which is nonempty for M large enough. We also point out that A =⋃

M>0EM . Letus fix M large enough so that EM 6= ∅. We want to construct φ ∈ X such that‖Dφ‖∞ ≤M and for all t ∈ EM ,

∫K(t)

c(x, t) divx φ(x, t) dx > 1/ε. In order to do so,we will need the notion of measurable selection. For more details than those givenbelow, we refer to [11].

We recall that if (Ω,A) is a measurable space, and Y is a complete separablemetric space, a map F : Ω → P(Y ) whose images are closed subsets of Y is said tobe measurable if for all open subset U ⊂ Y , F−1(U) = ω ∈ Ω; F (ω) ∩ U 6= ∅ ∈ A.A measurable map f : Ω → Y such that for all ω ∈ Ω, f(ω) ∈ F (ω), is called ameasurable selection of F .

If Z is another complete separable metric space, we say that g : Ω × Y → Z isCarathéodory if for all ω ∈ Ω, g(ω, ·) is continuous and if for all x ∈ Y , g(·, x) ismeasurable. In order to construct our function φ, we use the following measurableselection theorem:

Theorem 5.7 ([11], Theorem 8.2.9). Let (Ω,A) be a measurable space. Let Y andZ be two complete separable metric spaces, F : Ω → P(Y ) and G : Ω → P(Z) betwo measurable maps with closed images. Let g : Ω × Y 7→ Z be a Carathéodorymap. Then the map ω 7→ x ∈ F (ω); g(ω, x) ∈ G(ω) is measurable. Moreover if

∀ω ∈ Ω, g(ω, F (ω))∩G(ω) 6= ∅,

then there exists a measurable selection f of F such that g(ω, f(ω)) ∈ G(ω) for allω ∈ Ω.

73


We apply Theorem 5.7 with

Ω = EM , Z = R,

Y = φ ∈ C1(BR); ‖φ‖∞ ≤ 1 and ‖Dφ‖∞ ≤M with the norm ‖φ‖∞ + ‖Dφ‖∞,F (t) = φ ∈ Xt; ‖Dφ‖∞ ≤M, G(t) = [1/ε; +∞),

g(t, φ) =

∫

K(t)

c(x, t) div φ(x) dx.

We first notice that Y and Z are complete separable metric spaces, and that F andG have closed images. Moreover, G is constant, thus measurable.

1. For all t ∈ [0, T ],

F (t) = φ ∈ Y ; supx∈BR∩c(·,t)≥0|φ(x)| = 0,

and Theorem 5.7 applied to F1(t) = Y , G1(t) = 0, and

g1(t, φ) = supx∈BR∩c(·,t)≥0|φ(x)|,

shows that F is measurable on [0, T ]. Indeed g1 is Carathéodory: g1(·, φ) is clearlyusc, hence measurable.

2. Next we see that t 7→ g(t, φ) is measurable on [0, T ] for all φ ∈ C1(BR),because of Fubini’s theorem, since (x, t) 7→ 1K(t)(x) c(x, t) div φ(x) ∈ L1(RN×[0, T ]).Moreover φ 7→ g(t, φ) is continuous since the domain of integration is bounded andc is bounded, which shows that g is Carathéodory.

3. The supremum of measurable functions on a set that depends in a measurableway of t is measurable (cf [11, Theorem 8.2.11]), so Steps 1 and 2 show that

h : t 7→ supφ∈F (t)

∫

K(t)


is measurable on [0, T ]. But EM = h−1([1/ε; +∞))∩A, hence EM is Lebesgue mea-surable. By restriction, the maps F , G and g(·, φ) (for any φ ∈ Y ) are measurableon EM .

The assumptions of the measurable selection theorem are satisfied, and afterextending the measurable selection obtained off EM by multiplying it by 1EM

whichis measurable, we get the existence of φ ∈ X such that for all t ∈ EM ,

∫

K(t)

c(x, t) divx φ(x, t) dx ≥ 1/ε,

and this integral vanishes on [0, T ] \EM . Integrating this inequality between 0 andT , taking the supremum for φ ∈ X on the left-hand side, and letting M go toinfinity, we conclude, since A =

⋃M>0EM , that

supφ∈X

∫ T

0

∫

K(t)

c(x, t) divx φ(x, t) dxdt

≥ L(A)

ε.

74

6. Corresponding results for supersolutions

Since this is true for all ε > 0, and since L(A) > 0, sending ε to 0 contradicts (5.15)and concludes the proof of Lemma 5.5 .

Proof of Lemma 5.6. We start by noticing that the “≤” inequality of (5.19) isobvious. We know that there exists Ω of full measure in [0, T ] such that for all t ∈ Ω,

supφ∈Xθ

t

∫

∂∗K(t)

c−(x, t) 〈φ(x), νK(t)(x)〉 dHN−1(x)

< +∞.

Let ε > 0 be fixed. We apply Theorem 5.7 with this Ω and

Y = φ ∈ C0(BR); ‖φ‖∞ ≤ 1 with the norm ‖φ‖∞, Z = R,

F (t) = Xθt ,

G(t) = [ supφ∈Xθ

t

∫

∂∗K(t)

c−(x, t) 〈φ(x), νK(t)(x)〉 dHN−1(x)

− ε; +∞),

g(t, φ) =

∫

∂∗K(t)

c−(x, t) 1c<−θ(x, t) 〈φ(x), νK(t)(x)〉 dHN−1(x).

All the maps considered are measurable thanks to the previous arguments. Indeedg is clearly Carathéodory, since HN−1(c(·, t) < −θ ∩ ∂∗K(t)) < +∞, and anotheruse of Theorem 5.7 with Y1 = Z1 = R, F1(t) = R, G1(t) = [0,+∞), and

g1(t, y) = y − supφ∈Xθt

∫

∂∗K(t)

c−(x, t) 〈φ(x), νK(t)(x)〉 dHN−1(x)

+ ε

shows that G is measurable, which is the last missing verification.

As in the proof of Lemma 5.5, we obtain the existence of φ ∈ Xθ such that foralmost all t ∈ [0, T ],

∫

∂∗K(t)

c−(x, t) 1c<−θ(x, t) 〈φ(x, t), νK(t)(x)〉 dHN−1(x)

≥ supφ∈Xθ

t

∫

∂∗K(t)

c−(x, t) 1c<−θ(x, t) 〈φ(x), νK(t)(x)〉 dHN−1(x)

− ε.

Integrating this inequality between 0 and T , and then sending ε to 0 proves (5.19).

6 Corresponding results for supersolutions

We state in this section the counterparts for supersolutions of the main resultsproved up to now, and give the modifications needed for the proofs.

6.1 The integral formulation for supersolutions

Let c : RN × [0, T ] → R satisfy the assumptions of Theorem 2.2, and let themapping Ω : [0, T ] → P(RN) \ RN be such that

75


1. Ω is uniformly bounded on [0, T ], Ω(t) → Ω(0) as t → 0 and Ω(t) → Ω(T ) ast→ T in the Hausdorff distance.

2. The graph⋃

t∈[0,T ]t × Ω(t) of Ω is open in [0, T ] × RN .

3. u : (x, t) 7→ 1Ω(t)(x) is a viscosity supersolution of the eikonal equation

ut = c(x, t)|Du| in RN × (0, T ).

Set, for all ε > 0,Ωε(t) = x ∈ R

N ; dΩc(t)(x) > ε,and

cε(x, t) = min|y−x|≤ε

c(y, t).

Then we have:

Theorem 6.1. For all t1 and t2 satisfying 0 ≤ t1 ≤ t2 ≤ T , for almost all ε > 0,and for all φ ∈ C1(RN × [0, T ]; R+),

∫ t2

t1

∫

Ωε(t)

φt(x, t) dxdt+

∫ t2

t1

∫

dΩc(t)=εcε(x, t)φ(x, t) dHN−1(x)dt

≤[∫

Ωε(t)

φ(x, t) dx

]t2

t1

.

(6.1)

Remark 6.2. The fact that u is a supersolution of the equation does not guaran-tee, unlike the case of subsolutions, that the evolution is bounded. This is whyassumption 1 has to be stronger for supersolutions.

The only changes in the proof are:1. w(x, t) = dΩc(t)(x).2. We take θ : R → R non-decreasing of class C∞ such that θ = 0 in(−∞, 0], θ = 1 in [ε,+∞).3. The equivalent of Proposition 3.5 is proved by a regularization by inf-convolutioninstead of sup-convolution.

All arguments then follow in the same way.

6.2 The converse implication for supersolutions

The converse theorem for supersolutions naturally becomes:

Theorem 6.3. Let c : RN × [0, T ] → R satisfy the assumptions of Theorem 2.2 and

Ω : [0, T ] → P(RN) be such that

1. Ω is uniformly bounded on [0, T ].

2. The graph⋃


3. Inequality (6.1) holds for all 0 ≤ t1 ≤ t2 ≤ T , for almost all ε > 0 smallenough and for all φ ∈ C1(RN × [0, T ]; R+).

76

7. Global estimate

Then u : (x, t) 7→ 1Ω(t)(x) is a viscosity supersolution of ut = c(x, t)|Du| in RN ×

(0, T ).

In the proof of this result, the only change occurs when switching the integrationorder between x and σ (the equivalent of (4.4)), and we get:

∫ ε

0

∫

Ωσ(t)

φt(x, t) dxdσ =

∫

Ω(t)

dΩc(t)(x)φt(x, t) dx

which is exactly what we need to comply with the inequalities for supersolutions.

6.3 Regularity of the front

The analogue of Theorem 5.1 when we work on (6.1) is the following:

Theorem 6.4. Let c : RN × [0, T ] → R satisfy the assumptions of Theorem 2.2 and(A1), (A2). Let Ω : [0, T ] → P(RN ) be such that:

1. Ω is uniformly bounded on [0, T ], Ω(t) → Ω(0) as t → 0 and Ω(t) → Ω(T ) ast→ T in the Hausdorff distance.

2. The graph⋃


3. u : (x, t) 7→ 1Ω(t)(x) is a supersolution of the eikonal equation (3.1).

Then the following statements hold:(i) For a.e. t ∈ [0, T ], c(·, t)1Ω(t) has bounded variations in c(·, t) > 0.(ii) For a.e. t ∈ [0, T ], Ω(t) has locally finite perimeter in c(·, t) > 0.(iii) Denoting (·)+ the positive part of a quantity (x+ = max(x, 0)), we have:

∫ T

0

∫

∂∗Ω(t)

c+(x, t) dHN−1(x)dt < +∞,

an upper bound for this integral being given by

[∫

Ω(t)

1c>0(x, t) dx

]T

0

+

∫ T

0

∫

Ω(t)∩c(·,t)=0


The only modification is basically to switch all c < 0 to c > 0 and use themodified truncature Tη, a continuous non-decreasing and piecewise affine functionequal to 0 in (−∞, θ] and 1 in [θ + η,+∞) so as to isolate the set c > 0 insteadof c < 0.

7 Global estimate

We finally synthesize the results of Theorems 5.1 and 6.4 in order to get a globalestimate.

Theorem 7.1. Let c : RN × [0, T ] → R satisfy the assumptions of Theorem 2.2 and(A1), (A2), and let Ω : [0, T ] → P(RN ) and K : [0, T ] → P(RN )\∅ be such that:

77


1. K(0) is compact and for all t ∈ [0, T ], Ω(t) ⊂ K(t).

2. K(t) → K(0) and Ω(t) → Ω(0) in the Hausdorff distance as t→ 0,K(t) → K(T ) and Ω(t) → Ω(T ) in the Hausdorff distance as t→ T .

3. Graph(K) is closed in [0, T ]×RN and u : (x, t) 7→ 1K(t)(x) is a subsolution ofthe eikonal equation (3.1).

4. Graph(Ω) is open in [0, T ]×RN and u : (x, t) 7→ 1Ω(t)(x) is a supersolution ofthe eikonal equation (3.1).

5. LN+1(Graph(K) \Graph(Ω)) = 0.

Then for a.e. t ∈ [0, T ], K(t) has locally finite perimeter in c(·, t) 6= 0, and ifsgn(r) denotes the sign of r ∈ R (sgn(r) = r/|r| if r 6= 0, sgn(0) = 0), we have thefollowing estimate:

∫ T

0

∫

∂∗K(t)

|c(x, t)| dHN−1(x)dt

≤[∫

K(t)

sgn(c(x, t)) dx

]T

0

+ 2

∫ T

0

∫

K(t)∩c(·,t)=0


Remark 7.2. The existence of such Ω and K satisfying assumptions 1, 3 and 4 isensured by Theorem 2.2 and the paragraph that follows it. Assumption 2 avoidspathological behavior of the front at times 0 and T , and assumption 5 is related tothe so-called non-empty interior difficulty (see [22]).

Proof. Since LN+1(Graph(K) \ Graph(Ω)) = 0, we have for almost all t ∈ [0, T ],LN(K(t) \Ω(t)) = 0. Therefore K(t) has locally finite perimeter in c(·, t) 6= 0 foralmost all t such that K(t) has locally finite perimeter in c(·, t) < 0 and Ω(t) haslocally finite perimeter in c(·, t) > 0, and then ∂∗Ω(t) = ∂∗K(t). Finally, usingthe fact that |c| = c− + c+, we deduce the global estimate from the summation ofthe estimates of Theorems 5.1 and 6.4.

Acknowledgements. My grateful thanks go to Pierre Cardaliaguet for introduc-ing me to the subject of viscosity solutions, and for fruitful discussions during thepreparation of this work. I also thank Régis Monneau for pointing out that one ofthe original assumptions in Theorem 4.1 was not necessary.

78

Chapitre 2

Existence of weak solutions for

general nonlocal and nonlinear

second-order parabolic equations

Ce chapitre est issu de l’article [17]. Il s’agit d’un travail en collaboration avec G.Barles, P. Cardaliaguet et O. Ley.

On donne des résultats d’existence pour une classe générale d’équations paraboliquesnon-linéaires et non-locales du second ordre. La motivation principale de ce travailvient de la théorie des propagations de fronts dans le cas où la vitesse normaledépend du front de façon non-locale. Parmi les applications, nous considérons cer-taines équations apparaissant en théorie des dislocations, ainsi que dans l’étude dessystèmes de type FitzHugh-Nagumo.

Chapitre 2. Existence of weak solutions for general nonlocal equations

Abstract

In this article, we provide existence results for a general class of nonlocal

and nonlinear second-order parabolic equations. The main motivation

comes from front propagation theory in the cases when the normal veloc-

ity depends on the moving front in a nonlocal way. Among applications,

we present level-set equations appearing in dislocations’ theory and in

the study of FitzHugh-Nagumo systems.

Key words and phrases: Nonlocal Hamilton-Jacobi equa-tions, second-order equations, viscosity solutions, L1−dependencein time, front propagations, level-set approach, dislocation dynam-ics, FitzHugh-Nagumo system.

1 Introduction

We are concerned with a class of nonlocal and nonlinear parabolic equationswhich can be written as

ut = H [1u≥0](x, t, u,Du,D

2u) in RN × (0, T ),

u(·, 0) = u0 in RN ,(1.1)

where ut, Du and D2u stand respectively for the time derivative, gradient andHessian matrix with respect to the space variable x of u : RN × [0, T ] → R andwhere 1A denotes the indicator function of a set A. The initial datum u0 is abounded and Lipschitz continuous function on RN .

For any indicator function χ : RN × [0, T ] → 0, 1, or more generally for anyχ ∈ L∞(RN × [0, T ]; [0, 1]), H [χ] denotes a function of

(x, t, r, p, A) ∈ RN × [0, T ] × R × R

N \ 0 × SN ,

where SN is the set of real, N × N symmetric matrices. For almost any t ∈ [0, T ],(x, r, p, A) 7→ H [χ](x, t, r, p, A) is a continuous function on RN ×R×RN \ 0×SN

with a possible singularity at p = 0 (typically for geometrical equations, see forinstance [46]), while t 7→ H [χ](x, t, r, p, A) is a bounded measurable function for any(x, r, p, A) ∈ RN × R × RN \ 0 × SN .

We recall that the equation is said to be degenerate elliptic (or here parabolic)if, for any χ ∈ L∞(RN × [0, T ]; [0, 1]), for any (x, r, p) ∈ RN × R × RN \ 0, foralmost every t ∈ [0, T ] and for all A,B ∈ SN , one has

H [χ](x, t, r, p, A) ≤ H [χ](x, t, r, p, B) if A ≤ B,

where ≤ stands for the usual partial ordering for symmetric matrices.

Such equations arise typically when one aims at describing, through the so-called level-set approach, the motion of a family K(t)t∈[0,T ] of closed subsets ofRN evolving with a nonlocal velocity. Indeed, following the main idea of the level-setapproach, it is natural to introduce a function u such that

K(t) = x ∈ RN ; u(x, t) ≥ 0 ,

80

1. Introduction

and (1.1) can be seen as the level-set equation for u. In this framework, the non-linearity H corresponds to the velocity which, in the applications we have in mind,depends not only on the time, the position of the front, the normal direction and thecurvature matrix but also on nonlocal properties of the family K(t)t∈[0,T ] whichare carried by the dependence on 1u≥0. We may face rather different nonlocaldependences, and this is why we have chosen this formulation: in any case, theequation appears as a well-posed equation if we would consider the nonlocal depen-dence (i.e. 1u≥0) as being fixed; in other words the H [χ]-equation enjoys “good”properties.

Finally, we recall that, still in the case of level-set equations, the function u0 isused to represent the initial front, i.e.

u0 ≥ 0 = K0 and u0 = 0 = ∂K0 (1.2)

for some fixed compact set K0 ⊂ RN . We refer the reader to [46] and the referencestherein for precisions.

Let us turn to the main examples we have in mind.

1. Dislocation dynamics equations:

ut = (c0(·, t) ⋆ 1u(·,t)≥0(x) + c1(x, t))|Du|,

or ut =

[div

(Du

|Du|

)+ c0(·, t) ⋆ 1u(·,t)≥0(x) + c1(x, t)

]|Du|,

where

c0(·, t) ⋆ 1u(·,t)≥0(x) =

∫

RN

c0(x− y, t)1u(·,t)≥0(y)dy

and div (Du/|Du|) (x, t) is the mean curvature of the set u(·, t) = u(x, t) at x.Typically, the reasonable assumptions in this context (see, for example, [16]) are

the following: c0, c1 are bounded, continuous functions which are Lipschitz continu-ous in x (uniformly w.r.t. t) and c0, Dxc0 ∈ L∞([0, T ];L1(RN )). In particular, andthis is a key difference with the second example below, c0 is bounded.

2. FitzHugh-Nagumo type systems, which, in a simplified form, reduce to the non-local equation

ut = α

(∫ t

0

∫

RN

G(x− y, t− s) 1u≥0(y, s) dyds

)|Du|,

where α : R → R is Lipschitz continuous and G is the Green function of the heatequation (see (4.10)).

There are two key differences with the first example: the convolution kernel actsin space and time, and G is not bounded. This last difference plays a central rolewhen one tries to prove uniqueness (cf. Chapter 4, or [18]).

3. Equations of the form

ut = (k − LN (u(·, t) ≥ 0))|Du| (k ∈ R),

or ut =

[div

(Du

|Du|

)− LN(u(·, t) ≥ 0)

]|Du|,

81


where LN denotes the N -dimensional Lebesgue measure and therefore the velocityof the front at time t depends on the volume of K(t) = u(·, t) ≥ 0.

In the classical cases, the equations of the level-set approach are solved by usingthe theory of viscosity solutions. Nevertheless there are two key features which mayprevent a direct use of viscosity solutions’ theory to treat the above examples: themain problem is that these examples do not satisfy the right monotonicity property.This can be seen either through the fact that u ≥ 0 ⊂ v ≥ 0 does not implythat H [1u≥0] ≤ H [1v≥0], or by remarking that the associated front propagationsdo not satisfy the “inclusion principle” (geometrical monotonicity). Indeed, in thedislocation dynamics case, the kernel c0 changes sign, which implies the two abovefacts. Therefore the classical comparison arguments of viscosity solutions’ theoryfail, and since existence is also based on these arguments through Perron’s method,the existence of viscosity solutions to these equations becomes an issue too.

The second (and less important) feature which prevents a direct use of the stan-dard level-set approach arguments is the form of the nonlocal dependence on 1u≥0:as shown by Slepčev in [70] and used in the present framework (but in the monotonecase) in [16], a dependence in 1u≥u(x,t) is the most adapted to the level-set approachsince all the level sets of the solutions are treated similarly instead of having the0-level set playing a particular role.

As a consequence, we are going to use a notion of weak solution for (1.1) intro-duced in [16] (see Definition 2.1), and prove a general existence result. As a simpleapplication of this theorem, we recover existence results for dislocation equationsand the FitzHugh-Nagumo system obtained respectively in [16] and by Giga, Gotoand Ishii [48] or Soravia and Souganidis [72]. Let us mention that the technique ofproof of our results, using Kakutani’s fixed point theorem, is the same as the oneused in [48]. Here we generalize its range of application and combine it with a newstability result of Barles [15]. In Chapter 4, or [18], we prove the uniqueness of suchweak solutions for these two model equations in the case of positive velocities. Notethat the issue of uniqueness is a difficult problem and, in general, uniqueness doesnot hold as shown by the counterexample developed in [16].

Another issue of these nonlocal equations is the question of the behavior of Hwith respect to the size of the set u ≥ 0. Indeed, in the dislocation dynamics case,if c0 ∈ L∞([0, T ];L1(RN )), then H [1u≥0] is defined without restriction on the sizeof u ≥ 0. The situation is the same for the FitzHugh-Nagumo system. However,if c0 is only bounded and not in L1, or in volume-dependent equations, then we mayhave to impose that the support of u ≥ 0 be bounded for H [1u≥0] to be defined.This leads us to distinguish two cases, that we will call respectively the unboundedand the bounded case; we therefore prove two different existence results.

The chapter is organized as follows: in Section 2 we give a general definitionof a weak solution to (1.1), and state general assumptions on the Hamiltonians inSection 3. In Section 4 we prove existence of such solutions in the unbounded case,and apply our result to dislocation equations and the FitzHugh-Nagumo system. InSection 5 we treat the bounded case and give as an application an existence result

82

2. Definition of weak solutions

for volume-dependent equations.

Notation: In what follows, | · | denotes the standard Euclidean norm on RN or SN ,the space of real N ×N symmetric matrices. The notation B(x,R) (resp. B(x,R))represents the open (resp. closed) ball of radius R centered at x ∈ RN .

2 Definition of weak solutions

We will use the following definition of weak solutions introduced in [16]. To do so,we use the notion of viscosity solutions for equations with a measurable dependencein time which we call below “L1-viscosity solution”. We refer the reader to the generalintroduction for the definition of L1-viscosity solutions and [50, 61, 62, 27, 26] for acomplete presentation of the theory.

Definition 2.1. Let u : RN × [0, T ] → R be a continuous function. We say that uis a weak solution of (1.1) if there exists χ ∈ L∞(RN × [0, T ]; [0, 1]) such that

1. u is a L1-viscosity solution ofut(x, t) = H [χ](x, t, u,Du,D2u) in RN × (0, T ),u(·, 0) = u0 in RN .

(2.1)


1u(·,t)>0 ≤ χ(·, t) ≤ 1u(·,t)≥0 a.e. in RN . (2.2)


1u(·,t)>0 = 1u(·,t)≥0 a.e. in RN .

Remark 2.2. If for any fixed χ ∈ L∞(RN × [0, T ]; [0, 1]) the map H [χ] is geometric,then the map χ defined by (2.1)–(2.2) only depends on the 0-level-set of the initialcondition u0, as in the classical level-set approach. Indeed, let u1

0 : RN → R beanother bounded Lipschitz continuous map such that

u0 ≥ 0 = u10 ≥ 0 and u0 > 0 = u1

0 > 0,

and let u1 be the solution to (2.1) with the same χ but with initial condition u10

(under the assumptions of Theorem 4.1 or Theorem 5.1 such a solution exists and isunique). Then from the key property of geometric equations (see for instance [46])we will have, for almost all t ∈ [0, T ],

1u(·,t)>0 = 1u1(·,t)>0 ≤ χ(·, t) ≤ 1u(·,t)≥0 = 1u1(·,t)≥0 a.e. in RN .

This means that the map χ can be interpreted as the weak solution of a nonlocalgeometric flow.

83


3 Assumptions on the Hamiltonians

We first state some general assumptions which we use for both the unboundedand the bounded case. To simplify the presentation, we are going to formulateassumptions on the nonlinearities H [χ] which have to be satisfied for any χ ∈ X(and uniformly for such χ) where X is a subset of L∞(RN × [0, T ]; [0, 1]). We use adifferent X for the unbounded and for the bounded case.

(H1-X) Properties of the H [χ]−equation:

(i) For any χ ∈ X, Equation (2.1) has a bounded uniformly continuous L1-viscosity solution u : RN × [0, T ] → R. Moreover, there exists a constant L > 0independent of χ ∈ X such that ‖u‖∞ ≤ L.

(ii) For any fixed χ ∈ X, a comparison principle holds for Equation (2.1): if u is abounded, upper-semicontinuous L1-viscosity subsolution of (2.1) in RN×(0, T ) and vis a bounded, lower-semicontinuous L1-viscosity supersolution of (2.1) in R

N ×(0, T )with u(y, 0) ≤ v(y, 0) in RN , then u ≤ v in RN × [0, T ).

In the same manner, if u is a bounded, upper-semicontinuous L1-viscosity subso-lution of (2.1) in B(x,R)× (0, T ) for some x ∈ RN and R > 0, and v is a bounded,lower-semicontinuous L1-viscosity supersolution of (2.1) in B(x,R) × (0, T ) withu(y, t) ≤ v(y, t) if t = 0 or |y − x| = R, then u ≤ v in B(x,R) × [0, T ).

(H2-X) Regularity of H:

(i) For any compact subset K ⊂ RN ×R×RN \0×SN , there exists a boundedmodulus of continuity mK : [0, T ]×R+ → R+ such that mK(·, ε) → 0 in L1(0, T ) asε → 0, and

|H [χ](x1, t, r1, p1, A1) −H [χ](x2, t, r2, p2, A2)| ≤

mK(t, |x1 − x2| + |r1 − r2| + |p1 − p2| + |A1 − A2|)for any χ ∈ X, for almost all t ∈ [0, T ] and all (x1, r1, p1, A1), (x2, r2, p2, A2) ∈ K.

(ii) There exists a bounded function f(x, t, r), which is continuous in x and rfor almost every t and mesurable in t, such that: for any neighborhood V of (0, 0)in RN \ 0 × SN and any compact subset K ⊂ RN × R, there exists a boundedmodulus of continuity mK,V : [0, T ]×R+ → R+ such that mK,V (·, ε) → 0 in L1(0, T )as ε → 0, and

|H [χ](x, t, r, p, A) − f(x, t, r)| ≤ mK,V (t, |p| + |A|)

for any χ ∈ X, for almost all t ∈ [0, T ], all (x, r) ∈ K and (p, A) ∈ V .

(iii) If χn χ weakly-∗ in L∞(RN × [0, T ]; [0, 1]) with χn, χ ∈ X for all n, thenfor all (x, t, r, p, A) ∈ RN × [0, T ] × R × RN \ 0 × SN ,

∫ t

0

H [χn](x, s, r, p, A)ds −→n→+∞

∫ t

0

H [χ](x, s, r, p, A)ds

locally uniformly for t ∈ [0, T ].

84

4. Existence of weak solutions to (1.1) (unbounded case)

We have chosen to state Assumption (H1-X) in this form which may look ar-tificial: it means that we have existence, uniqueness of a continuous L1-viscositysolution u associated to any measurable fixed function 0 ≤ χ ≤ 1. For conditionson H under which (H1-X) is satisfied, we refer to [26, 62] and Section 4.2. More-over, (H1-X) (i) states that the u’s are bounded uniformly with respect to χ ∈ X:for the geometrical equations of the level-set approach, this uniform bound on u isautomatically satisfied with L = ‖u0‖∞ if (H1-X) (ii) holds, using that, in thiscase, constants are L1-viscosity solutions of (2.1). Assumption (H2-X) comes from[15] and will be used to apply a stability result for equations with L1-dependence intime.

4 Existence of weak solutions to (1.1) (unbounded

case)

In this section, we are interested in the case where the Hamiltonian H [χ] isdefined without any restriction on the size of the support of χ.

4.1 The existence theorem

Our general existence theorem is the following:

Theorem 4.1. Let u0 : RN → R be a bounded and Lipschitz continuous function.Assume that (H1-X) and (H2-X) hold with X = L∞(RN × [0, T ]; [0, 1]). Thenthere exists at least a weak solution to (1.1).

Remark 4.2. See also [15] for the stability of weak solutions.

Proof. From (H1-X), the set-valued mapping

ξ : X Xχ 7→

χ′ : 1u(·,t)>0 ≤ χ′(·, t) ≤ 1u(·,t)≥0 for almost all t ∈ [0, T ],where u is the L1-viscosity solution of (2.1)

,

is well-defined. Clearly, there exists a weak solution to (1.1) if there exists a fixedpoint χ of ξ, which means that χ ∈ ξ(χ). In this case the corresponding u is aweak solution of (1.1). We therefore aim at using Kakutani’s fixed point theoremfor set-valued mappings (see [10, Theorem 3 p. 232]).

In the Hausdorff convex space L∞(RN × [0, T ]; R) equipped with the L∞-weak-∗topology, the subset X is convex and compact (since it is closed and bounded for theL∞ norm). In the same way, for any χ ∈ X, ξ(χ) is a non-empty convex compactsubset of X for the L∞-weak-∗ topology.

Let us check that ξ is upper semicontinuous for this topology. It suffices to showthat, if

χn ∈ X L∞-weak-∗

χ and χ′n ∈ ξ(χn)

L∞-weak-∗χ′,

85


then

χ′ ∈ ξ(χ).

Let un be the unique L1-viscosity solution of (2.1) associated to χn by (H1-X).Using (H1-X), we know that the un’s are uniformly bounded. We can thereforedefine the half-relaxed limits

u = lim sup∗(un) and u = lim inf∗(un).

From (H2-X) (iii) (convergence of the Hamiltonians), we can apply Barles’ stabilityresult [15, Theorem 1.1] to obtain that u (respectively u) is a L1-viscosity subsolution(respectively supersolution) of (2.1) associated to χ.

In order to apply (H1-X) (ii) (comparison), we first have to show that

u(x, 0) ≤ u0(x) ≤ u(x, 0) in RN .

To do so, we recall that u0 is Lipschitz continuous, and therefore for all 0 < ε ≤ 1,we have, for any x, y ∈ RN ,

u0(y) ≤ u0(x) + ‖Du0‖∞|x− y| ≤ u0(x) + 2L|x− y|2ε2

+‖Du0‖2

∞ε2

8L, (4.1)

where L is the bound on ‖un‖∞ given by (H1-X) (i). We fix x and we argue in theball B(x, ε). From assumption (H2-X), we deduce that H [χ](x, t, r, p, A) is locallybounded, uniformly with respect to χ ∈ X. Therefore, the function

(y, t) 7→ ψε(y, t) = u0(x) + 2L|x− y|2ε2

+‖Du0‖2

∞ε2

8L+ Cεt

is a supersolution of the H [χn]-equation in the ball B(x, ε) provided that Cε is largeenough. Since un ≤ ψε on the parabolic boundary of B(x, ε)× [0, T ) thanks to (4.1),by (H1-X) (ii) (comparison), we obtain

un(y, t) ≤ ψε(y, t) in B(x, ε) × [0, T ) ,

and thenu(y, t) ≤ ψε(y, t) in B(x, ε) × [0, T ) .

Examining the right-hand side at (y, t) = (x, 0) and letting ε → 0 provides theinequality u(x, 0) ≤ u0(x). An analogous argument gives u(x, 0) ≥ u0(x).

Using again (H1-X) (ii) (comparison), we therefore have u ≤ u in RN × [0, T ),which implies that in RN × [0, T ), u = u coincide with the unique continuous L1-viscosity solution u of (2.1) associated to χ, as well as the local uniform convergenceof (un) to u.

Moreover, since χ′n ∈ ξ(χn), we have, for all ϕ ∈ L1(RN × [0, T ]; R+),

∫ T

0

∫

RN

ϕ 1un>0 ≤∫ T

0

∫

RN

ϕχ′n ≤

∫ T

0

∫

RN

ϕ 1un≥0.

86


Since χ′n

L∞-weak-∗χ′, applying Fatou’s Lemma, we get

∫ T

0

∫

RN

ϕ lim inf 1un>0 ≤∫ T

0

∫

RN

ϕχ′ ≤∫ T

0

∫

RN

ϕ lim sup 1un≥0.

But 1u>0 ≤ lim inf 1un>0 and lim sup 1un≥0 ≤ 1u≥0. It follows that

1u(·,t)>0 ≤ χ′ ≤ 1u(·,t)≥0 for a.e. t ∈ [0, T ],

and therefore χ′ ∈ ξ(χ).We infer the existence of a weak solution of (1.1) by Kakutani’s fixed point

theorem [10, Theorem 3 p. 232], as announced.

4.2 Applications

4.2.1 Dislocation dynamics equations

One important example for which Theorem 4.1 provides a weak solution is thedislocation dynamics equation (see the general introduction, or [5, 16] and the ref-erences therein), which reads

ut = [c0(·, t) ⋆ 1u(·,t)≥0(x) + c1(x, t)]|Du| in RN × (0, T ),u(·, 0) = u0 in RN ,

(4.2)

where

c0(·, t) ⋆ 1u(·,t)≥0(x) =

∫

RN

c0(x− y, t)1u(·,t)≥0(y) dy.

We assume that c0 and c1 satisfy the following assumptions:(D) (i) c0 ∈ C0([0, T ];L1

(RN)), c1 ∈ C0(RN × [0, T ]; R).

(ii) For any t ∈ [0, T ], c0(·, t) is locally Lipschitz continuous, and there exists aconstant C > 0 such that ‖Dc0‖L∞([0,T ];L1(RN )) ≤ C.

(iii) There exists a constant C such that, for any x, y ∈ RN and t ∈ [0, T ],

|c1(x, t)| ≤ C and |c1(x, t) − c1(y, t)| ≤ C|x− y|.

Theorem 4.3. Let u0 : RN → R be a bounded and Lipschitz continuous function.Under assumption (D), Equation (4.2) has at least a weak solution. Moreover, iffor all (x, χ) ∈ RN × L∞(RN × [0, T ]; [0, 1]) and almost all t ∈ [0, T ],

c0(·, t) ⋆ χ(·, t)(x) + c1(x, t) ≥ 0, (4.3)

and if there exists η > 0 with

|u0| + |Du0| ≥ η in RN in the viscosity sense, (4.4)

then any weak solution is classical.

87


Proof. This theorem is proved by Barles, Cardaliaguet, Ley and Monneau in [16,Theorem 1.2]. Another proof can be done using Theorem 4.1. First we note thatEquation (3.1) is a first-order particular case of (1.1) where H [χ] is defined, for all(x, p, χ) ∈ RN × RN × L∞(RN × [0, T ]; [0, 1]) and almost every t ∈ [0, T ], by

H [χ](x, t, p) = [c0(·, t) ⋆ χ(·, t)(x) + c1(x, t)]|p|.Assumption (H1-X) (i) is given by [16, Theorem 5.4] and [53] (for the regularitypart), while assumption (H1-X) (ii) holds thanks to the results of [62]. Assumption(H2-X) is given by [16, proof of Theorem 1.2]. It essentially amounts to noticingthat if χnχ in L∞-weak-∗, then by the definition of this convergence

∫ t

0

c0(·, s) ⋆ χn(·, s)(x) ds =

∫ t

0

∫

RN

c0(x− y, s)χn(y, s) dyds

→∫ t

0

∫

RN

c0(x− y, s)χ(y, s) dyds

=

∫ t

0

c0(·, s) ⋆ χ(·, s)(x) ds.

Finally, if (4.3) and (4.4) hold, the solutions are classical by [16, Theorem 1.3].

We can also consider the dislocation dynamics equation with an additional meancurvature term,

ut =

[div

(Du

|Du|

)+ c0(·, t) ⋆ 1u(·,t)≥0(x) + c1(x, t)

]|Du|, (4.5)

which has been studied by Forcadel and Monteillet for instance: see Chapter 3, or[44] . Theorem 4.1 also provides a weak solution to (4.5). In Chapter 3, however, westudy the problem with the particular tool of minimizing movements, which enablesus to construct a weak solution with χ of the particular form (x, t) 7→ 1E(t)(x),where E(t) ⊂ RN for any t ∈ [0, T ], the set-valued mapping t 7→ E(t) having goodregularity properties. This is due to the particular structure of (1.3), namely thepresence of the regularizing mean curvature term. Here we can deal with moregeneral nonlocal degenerate parabolic equations.

4.2.2 A FitzHugh-Nagumo type system

We are also interested in the following system mentioned in the general intro-duction,

ut = α(v)|Du| in RN × (0, T ),vt − ∆v = g+(v)1u≥0 + g−(v)(1 − 1u≥0) in R

N × (0, T ),u(·, 0) = u0, v(·, 0) = v0 in RN ,

(4.6)

which is obtained as the asymptotics as ε → 0 of the following FitzHugh-Nagumosystem arising in neural wave propagation or chemical kinetics (see [72]):

uε

t − ε∆uε = ε−1f(uε, vε),vε

t − ∆vε = g(uε, vε)(4.7)

88


in RN × (0, T ), where for (u, v) ∈ R

2,f(u, v) = u(1 − u)(u− a) − v (0 < a < 1),g(u, v) = u− γv (γ > 0).

The functions α, g+ and g− : R → R appearing in (4.6) are associated with f and g.This system has been studied in particular by Giga, Goto and Ishii [48] and Soravia,Souganidis [72]. They proved existence of a weak solution to (4.6). Here we recovertheir result as an application of Theorem 4.1.

If for χ ∈ L∞(RN × [0, T ]; [0, 1]), v denotes the solution ofvt − ∆v = g+(v)χ+ g−(v)(1 − χ) in RN × (0, T ),v(·, 0) = v0 in RN ,

(4.8)

and if c[χ](x, t) := α(v(x, t)), then Problem (4.6) reduces tout(x, t) = c[1u≥0](x, t)|Du(x, t)| in RN × (0, T ),u(·, 0) = u0 in RN ,

(4.9)

which is a particular case of (1.1). Let us now state the existence theorem of [48]and [72] that we can recover from our general existence theorem. We first gatherthe assumptions satisfied by α, g−, g+ and v0:

(F) (i) α is Lipschitz continuous on R.(ii) g+ and g− are Lipschitz continuous on RN , and there exist g and g in R

such thatg ≤ g−(r) ≤ g+(r) ≤ g for all r in R.

(iii) v0 is bounded and of class C1 with ‖Dv0‖∞ < +∞.

Theorem 4.4. Let u0 : RN → R be a bounded and Lipschitz continuous function.Under assumption (F), the problem (4.9), or equivalently the system (4.6), has atleast a weak solution. If in addition (4.4) holds and α ≥ 0, then any weak solutionis classical.

Proof. The explicit resolution of the inhomogeneous heat equation (4.8) shows thatfor any (x, t) ∈ RN × (0, T ),

v(x, t) =

∫

RN


+

∫ t

0

∫

RN



G(y, s) =1

(4πs)N/2e−

|y|2

4s . (4.10)

It is then easy to obtain the following lemma:

89


Lemma 4.5. Assume that g−, g+ and v0 satisfy (F). For χ ∈ L∞(RN×[0, T ]; [0, 1]),let v be the solution of (4.8). Set γ = max|g|, |g|. Then there exists a constant kN

depending only on N such that(i) v is uniformly bounded: for all (x, t) ∈ RN × [0, T ],

|v(x, t)| ≤ ‖v0‖∞ + γt.

(ii) v is continuous on RN × [0, T ].(iii) For any t ∈ [0, T ], v(·, t) is of class C1 in RN .(iv) For all t ∈ [0, T ], for all x, y ∈ RN ,

|v(x, t) − v(y, t)| ≤ ( ‖Dv0‖∞ + γkN

√t) |x− y|.

(v) For all 0 ≤ s ≤ t ≤ T, for all x ∈ RN ,

|v(x, t) − v(x, s)| ≤ kN(‖Dv0‖∞ + γkN

√s)

√t− s+ γ(t− s).

In particular the velocity c[χ] in (4.9) is bounded, continuous on RN × [0, T ] andLipschitz continuous in space, uniformly with respect to χ. From general resultson existence and comparison for classical viscosity solutions of the eikonal equationwith Lipschitz continuous initial datum (see for instance [16, Theorem 2.1]), weobtain that (H1-X) is satisfied.

Let us check (H2-X) (iii) ((i) and (ii) are straightforward): we claim that, ifχnχ in L∞-weak-∗, then for any x ∈ RN ,

∫ t

0

c[χn](x, s) ds→∫ t

0

c[χ](x, s) ds

locally uniformly in [0, T ]. Indeed, let vn (resp. v) be the solution of (4.8) with χn

(resp. χ) in the right-hand side. The estimates (iv) and (v) of Lemma 4.5 on theheat equation imply that we can extract by a diagonal argument a subsequence,still denoted (vn), which converges uniformly to some w in B(0, R) × [0, T ] for anyR > 0. We know that for any (x, t) ∈ RN × (0, T ),

vn(x, t) =

∫

RN


+

∫ t

0

∫

RN

G(x− y, t− s) [g+(vn)χn + g−(vn)(1 − χn)](y, s) dyds

where G is the Green function defined by (4.10). As n goes to infinity, we obtain

w(x, t) =

∫

RN


+

∫ t

0

∫

RN

G(x− y, t− s) [g+(w)χ+ g−(w)(1 − χ)](y, s) dyds.

90


Indeed∫ t

0

∫

RN

G(x− y, t− s) [g+(vn)χn + g−(vn)(1 − χn)](y, s) dyds

−∫ t

0

∫

RN

G(x− y, t− s) [g+(w)χ+ g−(w)(1 − χ)](y, s) dyds

=

∫ t

0

∫

RN

G(x− y, t− s) [(g+(w) − g−(w))(χn − χ)](y, s) dyds

+

∫ t

0

∫

RN

G(x− y, t− s)[χn (g+(vn) − g+(w)) +

(1 − χn) (g−(vn) − g−(w))](y, s) dyds.

The term∫ t

0

∫

RN

G(x− y, t− s) [(g+(w) − g−(w))(χn − χ)](y, s) dyds

converges to 0 since χnχ in L∞-weak-∗ and

|G(x− y, t− s) (g+(w) − g−(w))(y, s)| ≤ (g − g)G(x− y, t− s),

which is an integrable function of (y, s). The rest of the terms converges to 0 bydominated convergence since vn → w pointwise in RN × [0, T ] and

|χn (g+(vn) − g+(w)) + (1 − χn) (g−(vn) − g−(w))|≤ D |vn − w|≤ 2DM,

where D is a Lipschitz constant for g+ and g−, and M is a uniform bound for vn

and w given by Lemma 4.5 (i).This shows that w is the solution of (4.8), so that w = v. In particular (vn)

converges locally uniformly to v. We conclude that for any x ∈ RN ,

∫ t

0

c[χn](x, s) ds→∫ t

0

c[χ](x, s) ds

locally uniformly for t ∈ [0, T ] thanks to the Lipschitz continuity of α. This proves(H2-X) (iii), and we obtain existence of weak solutions to (4.9) according to The-orem 4.1.

If α ≥ 0 and (4.4) holds, then the fattening phenomenon for (2.1) does nothappen (see [22, 53]) so that, if u is any weak solution of (2.1), then for almost allt ∈ [0, T ],

1u(·,t)>0 = 1u(·,t)≥0 a.e. in RN ,

which means that u is a classical solution of (4.9). This completes the proof.

91


5 Existence of weak solution to (1.1) (bounded case)

It may happen that our Hamiltonian H [1u≥0] is only defined under some re-striction on the set u ≥ 0, for example if it remains bounded: this is typically thecase when a volume term is involved. For such cases, the existence of weak solutionsmay remain true, due to a particular framework.

5.1 The existence theorem

We use the following assumption:

(H4) There exists a bounded lower-semicontinuous function v : RN × [0, T ] → R

and R0 > 0 such that(i) v(x, t) < 0 if |x| ≥ R0, for any t ∈ [0, T ],(ii) v(x, 0) ≥ u0(x) in RN ,(iii) v is a supersolution of (2.1) for all χ ∈ X, where

X = χ ∈ L∞(RN × [0, T ]; [0, 1]); χ = 0 a.e. in v < 0 .

Assumption (H4) is some kind of compatibility condition between the equation andthe initial condition: of course, it implies that u0(x) < 0 if |x| ≥ R0 and, more orless, that the equation preserves this property (this is the meaning of v).

Under this assumption, we obtain the following existence result:

Theorem 5.1. Let u0 : RN → R be a bounded and Lipschitz continuous function.Assume (H4) and that (H1-X), (H2-X) and (H3-X) hold with X given in (H4).Then there exists at least a weak solution to (1.1).

Proof. The proof follows the arguments of the proof of Theorem 4.1: essentially, theonly change is the choice of X.

From (H4), the set-valued mapping

ξ : X Xχ 7→

χ′ : 1u(·,t)>0 ≤ χ′(·, t) ≤ 1u(·,t)≥0 for almost all t ∈ [0, T ],where u is the L1-viscosity solution of (2.1)

,

is well-defined: indeed, for any χ ∈ X, v is a supersolution of the H [χ]-equation andwe have u0(x) ≤ v(x, 0) in RN . Therefore, by (H1-X) (ii) (comparison), we have

u(x, t) ≤ v(x, t) in RN × [0, T ).

In particular, u(x, t) < 0 if v(x, t) < 0 and clearly any χ′ in ξ(χ) is in X. Weconclude exactly as in the proof of Theorem 4.1.

5.2 Applications

The typical cases we have in mind are geometrical equations; for instance,

ut =

[div

(Du

|Du|

)+ β

(LN(u(·, t) ≥ 0)

)]|Du| ,

92

5. Existence of weak solution to (1.1) (bounded case)

where β : R → R is a continuous function.In order to test condition (H4), it is natural to consider radially symmetric

supersolutions and typically, we look for supersolutions of the following form:

ψ(x, t) := R(t) − |x| ,

where R(·) is a C1-function of t. We point out two key arguments to justify thischoice: first ψ is concave in x, and checking the viscosity supersolution property isequivalent to checking it at points where ψ is smooth (because of the form of ψ).Next if ψ is a supersolution of the above pde, one can use (if necessary) a change offunction ψ → ϕ(ψ) with ϕ′ > 0, to ensure that ϕ(ψ) ≥ u0 in RN , and such that ϕis bounded, in order to use the comparison principle.

Plugging ψ in the equation, we obtain that ψ is a supersolution if

R′(t) ≥ −(N − 1)

|x| + β(ωNRN(t)) ,

where ωN = LN(B(0, 1)). The curvature term (N − 1)/|x| is not going to play anymajor role here since we are concerned with large R’s and the equation should holdon the 0-level set of ψ, i.e. for |x| = R.

Therefore, let us consider the ordinary differential equation

R′(t) = β(ωNRN(t)) with R(0) = R0 .

A natural condition for this equation to have solutions which do not blow up infinite time is the sublinearity in R of the right-hand side. This leads to the followingcondition on β:

β(t) ≤ L1 + L2 t1/N for any t > 0 ,

for some constants L1, L2 > 0. Under this condition, we easily build a function vsatisfying (H4). However if this condition is not satisfied, Theorem 5.1 only providesthe small time existence of solutions.

We complete this example by recalling that, for all χ ∈ L∞(RN × [0, T ]; [0, 1])with bounded support, we have a comparison result for the equation

ut =

[div

(Du

|Du|

)+ β

(∫

RN

χ(x, t)dx)

)]|Du| in R

N × (0, T )

(See Nunziante [61] and Bourgoing [27, 26]).

Aknowledgment. This work was partially supported by the ANR (Agence Na-tionale de la Recherche) through MICA project (ANR-06-BLAN-0082).

93

Chapitre 3

Minimizing movements for

dislocation dynamics with a mean

curvature term

Ce chapitre est issu de l’article [44] publié en ligne dans Control, Optimization andCalculus of Variations. Il s’agit d’un travail en collaboration avec N. Forcadel.

On y montre l’existence de mouvements minimisants pour la dynamique des disloca-tions avec un terme de courbure moyenne. On montre qu’un mouvement minimisantest une solution faible de cette loi d’évolution, dans un sens lié à la notion de so-lution de viscosité de l’équation level-set correspondante. On prouve également laconsistance de l’approche, en montrant que tout mouvement minimisant co¨avecl’évolution régulière tant que cette dernière existe. En relation avec cela, on montrefinalement l’existence en temps court d’une évolution régulière, à condition que laforme initiale soit suffisament régulière.

Chapitre 3. Minimizing movements for dislocations with mean curvature

Abstract

We prove existence of minimizing movements for the dislocation dynam-

ics evolution law with a mean curvature term. We prove that any such

minimizing movement is a weak solution of this evolution law, in a sense

related to viscosity solutions of the corresponding level-set equation. We

also prove the consistency of this approach, by showing that any min-

imizing movement coincides with the smooth evolution as long as the

latter exists. In relation with this, we finally prove short time existence

of a smooth front evolving according to our law, provided the initial

shape is smooth enough.

Key words and phrases: Front propagations, dislocation dy-namics, minimizing movements, sets of finite perimeter, currents,level-set approach, nonlocal Hamilton-Jacobi equations, viscositysolutions.

1 Introduction

In this chapter, we investigate the existence of minimizing movements (see Alm-gren, Taylor, Wang [1], Ambrosio [6], and the book by Ambrosio, Gigli and Savaré[9]) for a nonlocal geometric law governing the movement of a family K(t)0≤t≤T

of compact subsets of RN :

Vx,t = Hx,t + c0(·, t) ⋆ 1K(t)(x) + c1(x, t), (1.1)

where Vx,t denotes the normal velocity of ∂K(t) at a point x, Hx,t the mean curvatureof ∂K(t) at x (with negative sign for convex sets), ⋆ is the convolution in space,1K(t) is the indicator function of the set K(t) and c0, c1 : RN × [0, T ] → R are givenfunctions.

The nonlocal dependence c0(·, t) ⋆ 1K(t) in the expression of Vx,t is typical ofmodels for dislocation dynamics (see Alvarez, Hoch, Le Bouar and Monneau [5]).Moreover we think of the term c1 as a prescribed driving force. Equation (1.1) withonly these two terms (and without a mean curvature term) is currently also a centerof interest: in the context of viscosity solutions, its level-set formulation, namely

ut(x, t) = [c0(·, t) ⋆ 1u(·,t)≥0(x) + c1(x, t)]|Du(x, t)|, (1.2)

was first investigated by Alvarez, Hoch, Le Bouar and Monneau [5], who provedshort time existence and uniqueness of a viscosity solution to (1.2), and then byAlvarez, Cardaliaguet and Monneau [2], and by Barles and Ley [20], who proved, bydifferent methods, long time existence and uniqueness under suitable monotonicityassumptions. In (1.2) and throughout the chapter, ut denotes the time derivative ofu, Du denotes the space gradient of u, and | · | is the standard Euclidean norm. Themean curvature term in (1.1) corresponds to an additional line tension term in theelastic energy of the dislocation which better approximates what happens near thedislocation (see the introduction of [43] for a discussion on the model). The level-set

96

1. Introduction

formulation of the geometric law (1.1),

ut(x, t) =

[div

(Du

|Du|

)(x, t) + c0(·, t) ⋆ 1u(·,t)≥0(x) + c1(x, t)

]|Du(x, t)|, (1.3)

was studied by Forcadel in [43]. He proved short time existence and uniqueness ofa viscosity solution to (1.3).

In both cases, the source of major difficulties is the nonlocal dependence in theexpression of the velocity, c0(·, t) ⋆1K(t), which prevents the comparison principle tohold. Indeed, c0 is not necessarily non-negative, and physical models show that thissituation can not be avoided. The problem of existence and uniqueness of a viscositysolution to the level-set equations (1.2) and (1.3) for general kernels c0 is thereforestill open. For example, the long time existence and uniqueness results mentionedabove were obtained under the assumption that c0(·, t)⋆1E + c1(x, t) ≥ 0 for any setE, which guarantees that the dislocation is expanding, and a regularity assumptionon the initial shape K(0). The short time existence and uniqueness for (1.3) wasobtained in the case where the initial shape is a graph or a Lipschitz curve, withoutassumption on the sign of the nonlocal term. It is worth mentioning however thatthis equation benefits from the regularizing effect of the mean curvature term.

To overcome this difficulty, Barles, Cardaliaguet, Ley and Monneau defined in[16] a notion of weak solution for (1.2), and proved existence of such weak solu-tions under general assumptions on c0 and c1. A similar concept of solution hadalready been proposed by Giga, Goto and Ishii [72] for Fitzhugh-Nagumo systems.In Chapter 2, we investigated this notion of weak solution for general equations,including these two examples. In the present work, we wish to construct weak solu-tions for (1.1) of a special form, due to the regularizing mean curvature term. Wewill work with set-valued mappings E : [0, T ] → P(RN ) with uniformly boundedimages which are continuous in the L1 topology, that is to say, t 7→ 1E(t) belongs toC0([0, T ];L1(RN)). We assume that c0 and c1 satisfy some regularity assumptionswhich guarantee that (x, t) 7→ c0(·, t) ⋆ 1E(t)(x) + c1(x, t) is smooth enough for sucha mapping E. Let us now explain what we will call a weak solution of (1.1) in thischapter:

Definition 1.1 (Weak solutions).Assume that c0 ∈ Lip

([0, T ];L1(RN)

), c1 ∈ Lip

([0, T ];L∞(RN)

), that c0 and c1

are continuous on RN × [0, T ] and Lipschitz continuous in space. Let E : [0, T ] →P(RN ) be a set-valued mapping with uniformly bounded images such that t 7→ 1E(t)

belongs to C0([0, T ];L1(RN )).Let u : RN × [0, T ] → R be the unique uniformly continuous viscosity solution of

ut =

[div(

Du|Du|

)+ c0(·, t) ⋆ 1E(t)(x) + c1(x, t)

]|Du| for (x, t) ∈ RN × (0, T )

u(x, 0) = u0(x) for x ∈ RN ,

(1.4)

where u0 is a uniformly continuous function such that E0 = u0 ≥ 0,E0= u0 > 0.

97


We say that E is a weak solution of (1.1) if we have, for all t ∈ [0, T ],

u(·, t) > 0 ⊂ E(t) ⊂ u(·, t) ≥ 0 a.e. in RN .

We point out that this definition is slightly different from the definition of Chap-ter 2: indeed, what we called a weak solution there was the function u instead ofthe set E. In fact, this was justified by the fact that in general, we are only ableto build a solution u of (1.4) with 1E replaced by some χ ∈ L∞(RN × [0, T ]; [0, 1])which satisfies 1u>0 ≤ χ ≤ 1u≥0, but is not necessarily the indicator function ofsets. The situation is different here, due to the mean curvature term. Since in thischapter we adopt a very geometrical approach, and will work rather with sets thanfunctions, we temporarily use the above definition.

The goal of this chapter is to construct a weak solution to the geometric law (1.1).To do this, we wish to adapt the approach of Almgren, Taylor and Wang [1] (alsodiscovered independently by Luckhaus and Sturzenhecker [55])- initially proposedfor the mean curvature motion - to the geometric law (1.1) with its additionalnonlocal term and driving force. The idea of minimizing movements is, for a giveninitial set E0, to select a sequence of sets Eh(k) associated with time steps of sizeh by minimizing a suitable functional, so that the corresponding Euler equationis a discretization of our evolution law. A compactness result for sets of finiteperimeter guarantees the existence of a subsequence (hn) and a set-valued mappingE : [0, T ] 7→ P(RN ) such that Ehn([t/hn]) converges to E(t) in L1(RN) for allt, where [·] denotes the integer part. Such a E is called a minimizing movement(or generalized minimizing movement) associated to the geometric law. Moreover,we prove a priori estimates for the discrete evolution Eh, which imply the Höldercontinuity of the limit E in the appropriate metric. This guarantees that the setsE(t) cannot vary in a wildly discontinuous way.

Let us now explain the interest of this approach in the prospect of proving exis-tence of weak solutions. For any sequence (hn) going to 0 and such that Ehn([·/hn])converges to a minimizing movement E, we are able, thanks to the Euler equationcorresponding to our minimization procedure, to compute the velocity (in the vis-cosity sense) of the upper and lower limit of the Ehn(k)’s as n → +∞, E∗ and E∗,in function of E. This enables us to compare E∗ and E∗ with the 0-level set of theviscosity solution u appearing in Definition 1.1. Since E∗ ⊂ E ⊂ E∗, we will deducethat E is a weak solution of (1.1). In case no fattening occurs for u, we remark thatu is a viscosity solution of (1.3).

Of course it is a natural request that this construction be consistent with smoothflows if they exist. To verify this, we further show that if ∂E0 is a smooth hyper-surface, then there is a unique smooth solution for small times of the evolution law(1.1), and that any minimizing movement E coincides with this smooth evolutionas long as the latter exists. This uses the notions of lower/upper limits mentionedabove and of sub/super pairs of solutions of Cardaliaguet and Pasquignon [32].

To state our results in more details below, we first need to fix some notation andassumptions that will be used throughout the chapter.

98

1. Introduction

Notation

• For k ∈ N, Bkr (x) (resp. Bk

r (x)) denotes the open (resp. closed) ball of radius rcentered at x ∈ R

k, and Lk is the Lebesgue measure on Rk. If k is not specified, we

mean that k = N . We set ωk = Lk(Bk1 (0)). The Hausdorff measure of dimension k

on RN is denoted by Hk.

• The notation SN represents the set of real square symmetric matrices of size N .

• We say that a sequence (En) of subsets of RN converges to E in L1(RN) if 1En → 1E

in L1(RN) as n→ +∞.

• Let P be the set of all bounded subsets of RN having finite perimeter (see Chapter

1 or [38] for the definition and properties of sets of finite perimeter). We denote byP (E) the perimeter of E ∈ P, by P (E,U) the perimeter of E in U subset of RN ,and we endow P with the metric

δ(E,F ) = ‖1E − 1F‖L1(RN ) = LN(E∆F ),

where E∆F is the symmetric difference of E and F , i.e., E∆F = (E∪F )\ (E∩F ).In particular we call equivalent two sets E and F such that δ(E,F ) = 0, and we

also say that E = F almost everywhere (a.e.). Similarly, we say that E ⊂ F almosteverywhere if LN(E \ F ) = 0.

Moreover ∂∗E denotes the reduced boundary of E ∈ P. We also define a notionof boundary for E ∈ P that is invariant in the class of E formed by the sets thatare equivalent to E:

∂E = x ∈ RN ; 0 < LN(E ∩ Br(x)) < LN(Br(x)) for all r > 0.

Then ∂E is closed, and in fact ∂E = ∂∗E.

Definitions of tubes (see [30])

• For any subset E of [0, T ] × RN , we set E(t) = x ∈ RN ; (t, x) ∈ E. Converselya mapping t ∈ [0, T ] 7→ E(t) ∈ P(RN) can be seen as a subset of [0, T ] × RN byidentifying E with its graph ∪t∈[0,T ]t ×E(t).

• We call tube a bounded subset E of [0, T ] × RN . We call regular tube a tube Ewith C1 boundary in [0, T ] × R

N such that for any regular point (t, x) ∈ ∂E, theunit outer normal (νt, νx) to E at (t, x) satisfies νx 6= 0. In this case, the normalvelocity of E at (t, x) is −νt/|νx|.• Finally a mapping t ∈ [0, T ] 7→ Er(t) is said to be a smooth evolution with C3+α

boundary if Er is a compact regular tube such that Er(t) has C3+α boundary for allt ∈ [0, T ].

Assumptions on c0 and c1

Throughout the chapter, c0 and c1 are assumed to satisfy the following regularityassumption:

(A) c0 ∈ Lip([0, T ];L1(RN)

), c1 ∈ Lip

([0, T ];L∞(RN)

).

99


In particular, we set K0 = Lip(c0), and K1 = Lip(c1), so that for all t, s ∈ [0, T ],

‖c0(·, t) − c0(·, s)‖1 ≤ K0|t− s| and ‖c1(·, t) − c1(·, s)‖∞ ≤ K1|t− s|.We finally set

L0 = ‖c0‖L∞([0,T ];L1(RN )), L1 = ‖c1‖L∞([0,T ];L∞(RN )) and L = L0 + L1. (1.5)

We will sometimes need additional regularity for c0 and c1. When this happens,we will specify which assumptions are made in each of the statements of theorems.In particular we will sometimes need to require that c0 be symmetric, so that thegradient flow of our functional is, at least formally, a solution of (1.1):

(Symmetry of c0) We say that c0 is symmetric if c0(−(·), t) = c0(·, t) for allt ∈ [0, T ].

Main results

For h > 0 (the time step), k ∈ N such that kh ≤ T , E and F in P, we define,following the original idea of Almgren, Taylor and Wang [1], the functional

F(h, k, E, F ) = P (E)+1

h

∫

E∆F

d∂F (x) dx−∫

E

(1

2c0(·, kh) ⋆ 1E(x) + c1(x, kh)

)dx,

(1.6)where dC is the distance function to a closed set C.

Let us now define a minimizing movement:

Definition 1.2 (Minimizing movement [1]).Let T > 0 and E0 ∈ P. We say that E : [0, T ] → P is a minimizing movement

associated to F with initial condition E0 if there exist a sequence (hn), hn → 0+,and sets Ehn(k) ∈ P for all k ∈ N satisfying khn ≤ T , such that:

1. Ehn(0) = E0.

2. For any k, n ∈ N with (k + 1)hn ≤ T ,

Ehn(k + 1) minimizes the functional E → F(hn, k + 1, E, Ehn(k)) (1.7)

among all E ′s in P.

3. For any t ∈ [0, T ], Ehn([t/hn]) → E(t) in L1(RN) as n → +∞, where [·]denotes the integer part.

The first result of the chapter is the existence of minimizing movements associ-ated to our functional F :

Theorem 1.3 (Existence of minimizing movements).Assume that c0 and c1 satisfy (A). Let E0 ∈ P with LN(∂E0) = 0. Then, there

exists a minimizing movement E associated to F with initial condition E0 such thatfor all t, s satisfying t ≤ T and 0 ≤ s ≤ t < s+ 1, we have

δ(E(t), E(s)) ≤ γ (t− s)1

N+1 , (1.8)

where γ = γ(N, T,E0, K0, K1, L0, L1) is a constant.

100

1. Introduction

We then prove that any such minimizing movement is a weak solution of (1.1):

Theorem 1.4 (Minimizing movements are weak solutions).Assume that c0 is symmetric, that c0 and c1 satisfy (A), are continuous on

RN × [0, T ] and Lipschitz continuous in space. Let E0 ∈ P with LN(∂E0) = 0. Let

E be any minimizing movement associated to F with initial condition E0.

Then E is a weak solution of (1.1) in the sense of Definition 1.1. In particularif no fattening occurs, i.e. if the corresponding solution u of (1.4) is such thatu(·, t) = 0 has zero LN measure, then u is a viscosity solution of (1.3) with initialdatum u0.

Let us already point out that even in the absence of fattening (a favorable situ-ation which is not known to be generic), uniqueness for (1.3) is, to our knowledge,an open problem. The approach we use here provides one particular solution.

Our third result states that any minimizing movement E coincides with thesmooth evolution Er as long as the latter exists:

Theorem 1.5 (Agreement with the smooth flow).Assume that c0 is symmetric, that c0 and c1 satisfy (A), are continuous on

RN × [0, T ] and Lipschitz continuous in space. Let E0 be a compact subset of RN

with uniformly C3+α boundary. Let Er be a smooth evolution with C3+α boundarydefined on [0, T ], starting from E0, with normal velocity given by

Vx,t = Hx,t + c0(·, t) ⋆ 1Er(t)(x) + c1(x, t), (1.9)

where Hx,t is the mean curvature of ∂Er(t) at x.

Then any minimizing movement E associated to F with initial condition E0

satisfies E(t) = Er(t) almost everywhere, for all t ∈ [0, T ].

In relation with this, we finally prove short time existence and uniqueness of asmooth solution Er to (1.1), when E0 is sufficiently smooth. The regularity assump-tions on c0 and c1 are the following ones:

c0 ∈ L∞([0, T ];W 2,∞(RN )) ∩W 1,∞([0, T ];L∞(RN )) (1.10)

andc1 ∈W 2,1;∞(RN × [0, T ]), (1.11)

where f ∈W 1,∞([0, T ];L∞(RN)) means that f is Lipschitz continuous with respectto t ∈ [0, T ], uniformly with respect to x ∈ RN , and for n ∈ N∗,

W n,1;∞(RN × [0, T ]) =

f ∈ L∞(RN × [0, T ])

∣∣∣∣ft,

∂αf∂xα ∈ L∞(RN × [0, T ])

for α ∈ NN s.t.∑N

i=0 αi ≤ n

.

Theorem 1.6 (Existence and uniqueness of a smooth solution).Assume the regularity (1.10)-(1.11). Let E0 be a compact subset of RN with

uniformly C3+α boundary. Then there exists a small time t0 > 0 and a uniquesmooth evolution Er with C3+α boundary defined on [0, t0], starting from E0, withnormal velocity given by (1.9).

101


Let us now explain how this chapter is organized. First, in Section 2, we prove theexistence of minimizing movements and the Hölder estimate Theorem 1.3. Section3 is devoted to proving a regularity result for F -minimizers that we use in Section 4to prepare the proofs of Theorems 1.4 and 1.5, respectively given in Sections 5 and6. Finally, in Section 7, we prove Theorem 1.6.

2 Existence of minimizing movements

This section is concerned with the existence of minimizing movements associ-ated to F (Theorem 1.3). Let us start with existence and basic properties of F -minimizers.

2.1 F-minimizers

The first point to check is the existence of F -minimizers:

Proposition 2.1 (Existence of F -minimizers).For all h > 0, k ∈ N with kh ≤ T , and F ∈ P, there exists a minimizer of

E 7→ F(h, k, E, F ) on P. Moreover, if L is defined by (1.5), then

F ⊂ BR(0) a.e. ⇒ E ⊂ BR+Lh(0) a.e.

whenever E is a minimizer.

Proof. Let us fix F ∈ P with F ⊂ BR(0) a.e., and set B = BR+Lh(0). Let (En) bea minimizing sequence for F(h, k, ·, F ). We want to prove that for all n ∈ N,

F(h, k, En ∩ B,F ) ≤ F(h, k, En, F ). (2.1)

First, since B is open and convex, we know that

P (En ∩ B) ≤ P (En). (2.2)

Let us compare∫

Enc0(·, kh) ⋆ 1En(x) dx and

∫En∩B

c0(·, kh) ⋆ 1En∩B(x) dx: for allx ∈ RN ,

c0(·, kh) ⋆ 1En(x) =

∫

En

c0(x− y, kh) dy

= c0(·, kh) ⋆ 1En∩B(x) +

∫

En\Bc0(x− y, kh) dy.

Therefore∫

En

c0(·, kh) ⋆ 1En(x) dx =

∫

En∩B

c0(·, kh) ⋆ 1En∩B(x) dx

+

∫

En\Bc0(·, kh) ⋆ 1En∩B(x) dx+

∫

En

∫

En\Bc0(x− y, kh) dydx.

102

2. Existence of minimizing movements

Since ‖c0(·, kh) ⋆ 1A‖L∞(RN ) ≤ L0 for any measurable set A, it follows that∫

En

(1

2c0(·, kh) ⋆ 1En(x) + c1(x, kh)

)dx

≥∫

En∩B

(1

2c0(·, kh) ⋆ 1En∩B(x) + c1(x, kh)

)dx− LLN(En \B)

(2.3)

thanks to the definition of L (see (1.5)). Moreover F ⊂ B, so that

En∆F = (En ∩ B)∆F ∪ (En \B),

whence

1

h

∫

En∆F

d∂F (x) dx =1

h

∫

(En∩B)∆F

d∂F (x) dx+1

h

∫

En\Bd∂F (x) dx

≥ 1

h

∫

(En∩B)∆F

d∂F (x) dx+ LLN(En \B),

(2.4)

since d∂F (x) ≥ Lh for all x ∈ En\B by definition of B. Putting (2.2), (2.3) and (2.4)together proves (2.1). Therefore we can replace (En) by (En ∩ B) as a minimizingsequence, and in particular we can assume that En ⊂ B for all n. Then

F(h, k, En, F ) ≥ −∫

En

(1

2c0(·, kh) ⋆ 1En(x) + c1(x, kh)

)dx

≥ −(

1

2L0 + L1

)LN(B),

so that infE∈P

F(h, k, E, F ) > −∞. Besides, for n large enough,

F(h, k, En, F ) ≤ infE∈P

F(h, k, E, F ) + 1.

This implies that

P (En) ≤ infE∈P

F(h, k, E, F ) + 1 +

(1

2L0 + L1

)LN(B),

and gives a uniform bound on the perimeter of the En’s. Since they are also uni-formly bounded by B, it follows from the compactness theorem for sets of finiteperimeter [38, Section 5.2.3] that we can extract a converging subsequence (Enk

)of (En) in the sense that there exists E∞ ∈ P, E∞ ⊂ B, such that Enk

→ E∞ inL1(RN). Therefore

F(h, k, E∞, F ) ≤ lim infk→∞

F(h, k, Enk, F ) = inf

E∈PF(h, k, E, F ),

because all terms in the expression of F are at least lower semi-continuous in theE variable for the L1 topology. Thus E∞ is a minimizer of E 7→ F(h, k, E, F ) onP. Finally, if E is any other minimizer, then the previous comparisons show thatP (E ∩B) = P (E), whence E ⊂ B almost everywhere (see the comparison theorem[6, p. 216]).

103


Remark 2.2. This proposition shows that the Eh(k)’s are uniformly bounded forall h and k, if E0 ∈ P: more precisely, if E0 ⊂ BR(0) a.e., then since kh ≤ T ,we can choose Eh(k) ⊂ BR+LT (0) independently of k, h. Therefore we can chooseΩ = BR+LT+1(0) so that Eh(k) ⋐ Ω for all k, h. We will always do so in this chapter,and set D = R + LT + 1.

Remark 2.2 gives a uniform bound Ω for Eh(k), independently of h, k, providedthat E0 is bounded. In order to have compactness in P, so as to construct ourminimizing movement, we also want a uniform bound on the perimeter of Eh(k).

Proposition 2.3 (Uniform bound on the perimeter).Let E0 ∈ P with E0 ⊂ BR(0). Then, there exists a constant c > 0 depending on

T,E0, D, L0, L1, K0, K1 but independent of h and k such that if Eh is defined by theprocedure (1.7), we have

P (Eh(k)) ≤ c for allh, k such that kh ≤ T.

Proof. By definition of Eh, for any j ≥ 1 such that jh ≤ T , we have

F(h, j, Eh(j), Eh(j − 1)) ≤ F(h, j, Eh(j − 1), Eh(j − 1)),

and in particular,

P (Eh(j)) −∫

Eh(j)

(1

2c0(·, jh) ⋆ 1Eh(j)(x) + c1(x, jh)

)dx

≤ P (Eh(j − 1)) −∫

Eh(j−1)

(1

2c0(·, jh) ⋆ 1Eh(j−1)(x) + c1(x, jh)

)dx.

Adding these inequalities for j = 1, . . . , k with kh ≤ T , we find

P (Eh(k)) − P (E0) ≤k∑

j=1

Jh(j, j) − Jh(j − 1, j)

=k∑

j=1

∫

Ω

c1(·, jh)1Eh(j) − c1(·, jh)1Eh(j−1)

+1

2

k∑

j=1

∫

Ω

(c0(·, jh) ⋆ 1Eh(j))1Eh(j) − (c0(·, jh) ⋆ 1Eh(j−1))1Eh(j−1),

(2.5)

where we have set

Jh(i, j) =

∫

Eh(i)

(1

2c0(·, jh) ⋆ 1Eh(i)(x) + c1(x, jh)

)dx. (2.6)

104


Doing an Abel transformation on the first sum of the last member of (2.5) yields

k∑

j=1

∫

Ω

c1(·, jh)1Eh(j) − c1(·, jh)1Eh(j−1)

=

∫

Ω

c1(·, kh)1Eh(k) −∫

Ω

c1(·, h)1E0 +

k−1∑

j=1

∫

Ω

[c1(·, jh) − c1(·, (j + 1)h)]1Eh(j)

≤ 2L1 LN(Ω) + (k − 1)K1 hLN(Ω)

≤ (2L1 +K1 T )LN(Ω).

The same manipulation with the second sum gives

k∑

j=1

∫

Ω

(c0(·, jh) ⋆ 1Eh(j))1Eh(j) − (c0(·, jh) ⋆ 1Eh(j−1))1Eh(j−1)

≤ (2L0 +K0 T )LN(Ω).

This proves that for all k such that kh ≤ T ,

k∑

j=1

Jh(j, j) − Jh(j − 1, j) ≤ (L0 + 2L1 +1

2K0 T +K1 T )LN(Ω) (2.7)

and gives the result, with c = P (E0) + (L0 + 2L1 + 12K0 T +K1 T )LN(Ω).

2.2 Minimizing movements

We are now ready to address the problem of existence of minimizing movements.Proofs in this section closely follow the ideas of Almgren, Taylor and Wang [1], andare adaptations of Ambrosio’s simplified presentation of these ideas (see [6]).

The main result in the prospect of the proof of existence of minimizing movementsis the following theorem on the behaviour of the solutions of procedure (1.7):

Theorem 2.4 (Discrete Hölder estimate).Let E0 ∈ P with E0 ⊂ BR(0). There exists a constant γ = γ(N,D) > 0 (where

D is defined in Remark 2.2) and h0 > 0 such that for all h ∈ (0, h0), for all m, l ∈ N

satisfying mh ≤ T and 0 < l < m < l + 1h, we have

δ(Eh(m), Eh(l)) ≤ γ c [h(m− l)]1

N+1 , (2.8)

where c is the uniform bound on P (Eh(k)) given by Proposition 2.3.

Theorem 1.3 is a corollary of this result, as proved in [6, pp. 231-232]. Howeverthe arguments of [1, Theorem 4.4] or [6, Theorem 3.3] for the proof of Theorem 2.4in the mean curvature motion case need a few adaptations due to the particularform of F . This is what the rest of this section is devoted to. We begin by givingsome preliminary results which will be necessary in the proof of Theorem 2.4.

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Lower density bound for F-minimizers

Theorem 2.5 (Lower density bound for F -minimizers).There exist two positive constants α and β (depending only on N) and h0 > 0

such that if E ∈ P is a minimizer of F(h, k, ·, F ) with F ∈ P, E ∪ F ⊂ BD−1(0),and h ∈ (0, h0), then

∀x ∈ ∂E, ∀ρ ∈(

0,αh

D

), P (E,Bρ(x)) ≥ βρN−1. (2.9)

Proof. The proof relies on the following lemma relating the perimeter of E ∈ P andthe perimeter of E replaced by a cone in a small ball:

Lemma 2.6 ([6], Lemma 3.5). Let E ∈ P, x ∈ RN and f(ρ) = P (E,Bρ(x)). Set

Eρ = (E ∩ (RN \Bρ(x))) ∪y ∈ Bρ(x); x+ ρ

y − x

|y − x| ∈ E

.

Then for almost all ρ > 0 (all ρ such that f is differentiable at ρ), we have

P (Eρ, Bρ(x)) ≤ ρf ′(ρ)

N − 1.

Let us now prove Theorem 2.5. Fix x ∈ ∂∗E and ρ > 0 such that f is differen-tiable at ρ. By definition of E, we know that F(h, k, E, F ) ≤ F(h, k, Eρ, F ), that isto say

P (E) ≤ P (Eρ) +1

h

∫

Eρ∆F

d∂F (y) dy −∫

E∆F

d∂F (y) dy

(2.10)

+

∫

E

(1

2c0(·, kh) ⋆ 1E(y) + c1(y, kh)

)dy −

∫

Eρ

(1

2c0(·, kh) ⋆ 1Eρ(y) + c1(y, kh)

)dy.

But since E coincides with Eρ in RN \ Bρ, we have

P (E,RN \ Bρ(x)) = P (Eρ,RN \ Bρ(x)). (2.11)

Moreover f is continuous at ρ, which together with (2.10) and (2.11) implies that

P (E,Bρ(x)) = P (E, Bρ(x)) ≤ P (Eρ, Bρ(x)) +2D

hωNρ

N + 2LωNρN ,

due to the fact that d∂F (y) ≤ 2D for all y ∈ Bρ(x), provided ρ < 1. Now Lemma2.6 implies that for almost all ρ ∈ (0, 1),

f(ρ) ≤ ρf ′(ρ)

N − 1+

(2D

h+ 2L

)ωNρ

N . (2.12)

Therefore, the function

g : ρ 7→ f(ρ)

ρN−1+

(2D

h+ 2L

)(N − 1)ωN ρ

106


is non-decreasing on (0, 1). In particular if x ∈ ∂∗E and ρ ∈ (0, 1),

g(ρ) ≥ lim infρ→0+

g(ρ) ≥ ωN−1 (2.13)

because of [38, Corollary 1 (ii) p. 203]. As a consequence, for all ρ ∈ (0, 1),

f(ρ) ≥ ωN−1ρN−1 −

(2D

h+ 2L

)(N − 1)ωN ρ

N . (2.14)

Let us set α = ωN−1

8(N−1)ωNand β = ωN−1

2. Then, provided h < minD

L, D

α =: h0, we

deduce from (2.14) that for all ρ ∈ (0, αhD

),

P (E,Bρ(x)) = f(ρ) ≥ β ρN−1.

Since ∂∗E is dense in ∂E, this also holds for all x ∈ ∂E.

Corollary 2.7 ([6], Corollary 3.6). Let E ∈ P be a minimizer of F(h, k, ·, F ) withF ∈ P and h ∈ (0, h0). Then

HN−1(∂E \ ∂∗E) = 0.

Distance-Volume comparison

We recall here a general result which makes it possible to compare LN(A \ C) and∫Ad∂C under conditions of density of C similar to (2.9). Such comparison will be

essential to prove Theorem 2.4.

Theorem 2.8 (Distance-Volume comparison, [6], p. 230).Let C be a compact subset of RN such that there exist β > 0 and τ > 0 with

HN−1(C ∩Bρ(x)) ≥ βρN−1 for any x ∈ ∂C and ρ ∈ (0, τ).

Then there exists a constant Γ = Γ(N) > 0 such that for all R > τ , for all Borel setA ⊂ R

N , we have

LN(A \ C) ≤[2Γ

(R

τ

)N−1

HN−1(C)

] 12 [∫

A

dC(x) dx

] 12

+1

R

∫

A

dC(x) dx. (2.15)

We are now able to prove Theorem 2.4.

Proof of Theorem 2.4. Let us fix h ∈ (0, h0), where h0 is given by Theorem 2.5. Bydefinition of Eh, we have for any j ≥ 1 such that jh ≤ T ,

F(h, j, Eh(j), Eh(j − 1)) ≤ F(h, j, Eh(j − 1), Eh(j − 1)),

that is to say,∫

Eh(j)∆Eh(j−1)

d∂Eh(j−1)(x) dx ≤ h [P (Eh(j−1))−P (Eh(j))]+ h [Jh(j, j)−Jh(j−1, j)],

107


where Jh(i, j) is defined by (2.6). Let us set

Ih(j) = [P (Eh(j − 1)) − P (Eh(j))] + [Jh(j, j) − Jh(j − 1, j)] 12 .

We now use Theorem 2.8 with C = ∂Eh(j−1), A = Eh(j)∆Eh(j−1), τ = αhD

, whichis justified for j ≥ 2 because of the density estimate (2.9). Thanks to Corollary 2.7,we know that LN(C) = 0, so that for all R > αh

D,

LN(Eh(j)∆Eh(j − 1)) (2.16)

≤[2Γ

(R

τ

)N−1

HN−1(∂Eh(j − 1))

]12 √

h Ih(j) +1

Rh Ih(j)

2.

Recall that Proposition 2.3 gives a uniform bound c on the perimeter of F -minimizers,so that HN−1(∂Eh(j − 1)) ≤ c.

Let m, l ∈ N satisfy mh ≤ T and 0 < l < m < l + 1h. We choose

R =αh

D[h(m− l)]

−1N+1 >

αh

D,

and add up inequalities (2.16) for j = l+1, . . . , m. Recall that (2.5) and (2.7) showthat

m∑

j=l+1

Ih(j)2 ≤P (Eh(l)) +

m∑

j=l+1

Jh(j, j) − Jh(j − 1, j)

≤P (E0) +

m∑

j=1

Jh(j, j) − Jh(j − 1, j) ≤ c.

Moreover, the Cauchy-Schwarz inequality shows that

m∑

j=l+1

Ih(j) ≤√m− l

m∑

j=l+1

Ih(j)2

12

≤√m− l

√c.

Finally, we find that

LN(Eh(m)∆Eh(l)) ≤[2Γ[h(m− l)]−

N−1N+1 c

] 12√h(m− l)

√c+

D

αh[h(m− l)]

1N+1 h c

=

(√2Γ +

D

α

)c [h(m− l)]

1N+1 ,

which concludes the proof.

3 Regularity for F-minimizers

One of the main interests of the variational approach used in [1] is that it enablesto use the regularity theory for area-minimizing currents described for instance in

108

3. Regularity for F-minimizers

[25, 41, 60, 69]. This is the idea we follow in this section. We use the notation of [1].In particular, the notation M and S stand respectively for the mass and size of anintegral current: if T is a k integral current associated to a k rectifiable set S ⊂ RN

and a density function θ, then M(T ) =∫

Sθ dHk, while S(T ) = Hk(S) (see [1, §

3.1.3]). Besides, if E ∈ P, [E] denotes the solid associated to E, i.e. the canonicalN -dimensional Euclidean current restricted to E. We use the notation T |LC forthe restriction of a current T to a set C.

3.1 Existence of tangent cones

A fundamental notion in regularity theory is that of tangent cones defined asfollows:

Definition 3.1. Let fp,R : x 7→ R(x − p), for p ∈ RN , R > 0. A locally integralcurrent [J ] is called a tangent current to ∂E at p ∈ ∂E if there exists a sequence(Ri) converging to +∞ such that if we set E(R) = fp,R(E), then [E(Ri)] → [J ]locally as i→ +∞, in the sense that LN((J∆E(Ri)) ∩Br(q)) → 0 for each q ∈ RN

and r > 0.

Lemma 3.2 (Existence of tangent cones).Let F ∈ P and let E be a minimizer of F(h, k, ·, F ) on P. For each p ∈ ∂E,

there exists a tangent current [J ] to ∂E at p. Each such tangent current [J ] is acone and locally minimizes the perimeter P . Moreover 0 ∈ ∂J .

Proof. The proof is inspired by that of [1, Theorem 3.9]. We easily check that forall R > 0,

P (E(R)) = RN−1P (E),

1

h

∫

E(R)∆F (R)

d∂F (R)(y) dy = RN+1 1

h

∫

E∆F

d∂F (y) dy,

∫

E(R)

1

2c0(·, kh) ⋆ 1E(R)(y) dy = R2N

∫

E

1

2c0(R(·), kh) ⋆ 1E(y) dy

∫

E(R)

c1(y, kh) dy = RN

∫

E

c1(R(y − p), kh) dy.

By definition of E we find that E(R) minimizes

E 7→P (E) +1

R2h

∫

E∆F (R)

d∂F (R)(y) dy

− 1

RN+1

∫

E

1

2cR0 (·, kh) ⋆ 1E(y) dy − 1

R

∫

E

cR1 (y, kh) dy, (3.1)

where we have set cR0 (y, t) = c0(y/R, t), cR1 (y, t) = c1(p + y/R, t). Let us compareE(R) and E(R) \ Br(q) for fixed q ∈ RN and r > 0, with respect to this lastfunctional. It follows from manipulations similar to those in the proof of Proposition2.1 that for almost all r > 0,

P (E(R), Br(q)) ≤ P (Br(q)) +1

R2h

∫

Br(q)

d∂F (R)(x) dx+L

RLN(Br(q)),

109


where L is defined by (1.5). But diamF (R) = RdiamF , so that

R 7→ 1

R2h

∫

Br(q)

d∂F (R)(x) dx

is bounded as a function of R, and even converges to 0 as R goes to infinity. Thisprovides the sufficient bound on the perimeter of E(R) in balls to infer the existenceof a tangent current [J ] (using the compactness result [67, Theorem 1.1 p. 225]).

Let us prove that [J ] locally minimizes the perimeter. This means that for allq ∈ RN , all r > 0, and all (N − 1) integral currents X with ∂X = 0 and havingsupport in C = Br(q), then M(∂[J ] |LC) ≤ M(∂[J ] |LC +X). We first recall from[1, § 3.1.6] that there exists an N integral current Q with compact support in Csuch that ∂Q = X and

S(Q) ≤ M(Q) ≤ r

NM(X). (3.2)

Then according to [41, § 4.5.17], we can write Q as

Q =

+∞∑

i=0

[Qi] −+∞∑

i=0

[Pi],

where Qi, Pi ∈ P and (Qi), (Pi) are nested families such that P1 ∪ Q1 ⊂ Supp(Q)and P1 ∩Q1 = ∅. Let us set K = (E(R) \ P1) ∪Q1 and compare E(R) and K withrespect to the functional defined by (3.1). We obtain that

P (E(R)) ≤ P (K) +1

R2h

∫

P1∪Q1

d∂F (R)(x) dx+L

RLN(P1 ∪Q1)

≤ M(∂[E(R)] + ∂Q) +1

R2h

∫

C

d∂F (R)(x) dx+L

RS(Q).

Since Q and ∂Q = X have compact support in C, and since

P (E(R), C) = M(∂[E(R)] |LC),

we deduce that

M(∂[E(R)] |LC) ≤ M(∂[E(R)] |LC +X) +1

R2h

∫

C

d∂F (R)(x) dx+L

RLN(C).

Knowing this, we can adapt Theorem 34.5 of [69] to show that [J ] locally minimizesthe perimeter and also that if Ri is such that [E(Ri)] → [J ] as i → +∞, then forall x ∈ RN and almost all ρ > 0, P (E(Ri), Bρ(x)) → P (J,Bρ(x)) as i→ +∞.

Finally we check that [J ] is a cone, i.e. that J is invariant under all homotheticexpansions z 7→ λz for λ > 0. To see this we recall from (2.12) and (2.13) that forall x ∈ ∂E, the function

g : ρ 7→ P (E,Bρ(x))

ρN−1+ cρ

110

3. Regularity for F-minimizers

is non-decreasing on (0, 1), where c is a constant, and that for all ρ ∈ (0, 1),

P (E,Bρ(x))

ρN−1+ cρ ≥ ωN−1.

It follows that ∂E has a density θ(∂E, x) at x with θ(∂E, x) ≥ 1. For all ρ > 0,

P (E(R), Bρ(0))

ρN−1=P (E,Bρ/R(p))

(ρ/R)N−1−→

R→+∞θ(∂E, p)ωN−1.

Moreover for almost all ρ > 0,

P (E(Ri), Bρ(0))

ρN−1−→

i→+∞

P (J,Bρ(0))

ρN−1.

This shows that the ratio ρ1−NP (J,Bρ(0)) is independent of ρ, which is known toimply that J is a cone (see the proof of Theorem 9.3 in [49]). In fact, we know thatρ1−NP (J,Bρ(0)) = θ(∂E, p)ωN−1 > 0, so that 0 ∈ ∂J and θ(∂J, 0) = θ(∂E, p).

3.2 The regularity results

The existence of tangent cones enables us to prove regularity results for F -minimizers, as in [1, § 3.5 and 3.7].

Theorem 3.3 (C1-regularity for F -minimizers).Let F ∈ P, and let E be a minimizer of F(h, k, ·, F ) on P. Then ∂E is a

C1-hypersurface, except for a set of Hausdorff dimension less than N − 8 (empty ifN ≤ 7).

Proof. We check that E is an almost minimal current in the sense of Bombieri, thatis, for some δ > 0, for all (N − 1) integral current X with ∂X = 0 and havingcompact support in C with diam(C) = r ≤ δ, then

M(∂[E] |LC) ≤ (1 + ω(r))M(∂[E] |LC +X) (3.3)

for some function ω such that ω(r) → 0 as r → 0+. To do so we proceed as in theprevious proof, write X = ∂Q with

Q =

+∞∑

i=0

[Qi] −+∞∑

i=0

[Pi],

set K = (E \ P1) ∪Q1 and compare E and K with respect to F :

P (E,C) ≤ P (K) +1

h

∫

P1∪Q1

d∂F (y) dy + LLN (P1 ∪Q1).

111


Let D > 0 be such that E ∪ F ⊂ BD−1(0). If BD−1(0) ∩ C 6= ∅ (otherwise (3.3) isobvious), and δ ≤ 1, the previous comparison yields

M(∂[E] |LC)

≤ M(∂[E] |LC + ∂Q) +

(2D

h+ L

)S(Q)

≤ M(∂[E] |LC +X) +

(2D

h+ L

)r

NM(X) (using (3.2))

≤ M(∂[E] |LC +X) +

(2D

h+ L

)r

N(M(∂[E] |LC +X) + M(∂[E] |LC)).

This easily implies the result with ω(r) = 3(

2Dh

+ L)

rN

and δ = N3

(2Dh

+ L)−1

.

In addition, at any point p of ∂E there exists a tangent cone [J ] which minimizesthe perimeter (by Lemma 3.2). Such a cone must be a hyperplane for N ≤ 7 (see[69, Appendix B]), so that in particular θ(E, p) = θ(J, 0) = 1. We then deduce theresult from the final remark in [25]. In case N ≥ 8, we use the dimension reductionargument of Federer (see [49, Theorem 11.8]).

Now, we prove that minimizers are smooth at contact points with smooth hy-persurfaces:

Theorem 3.4. Let F ∈ P, and let E be a minimizer of F(h, k, ·, F ) on P. Assumethat there exists a closed set K ⊂ R

N such that ∂K is a C1 hypersurface and∂E ∩K = p. Then ∂E is a C1 hypersurface near p.

Proof. Let [J ] be any tangent cone to ∂E at p. The assumption that ∂E ∩K = pguarantees that ∂J is contained in the closed half-space orthogonal to the outer unitnormal n to K at p and containing n. Since 0 ∈ ∂J , Theorem 15.5 p. 174 of [49]implies that ∂J is regular at 0, and therefore is a hyperplane. The result follows asin the proof of Theorem 3.3.

Actually, we can deduce higher regularity for F -minimizers at each point wherethey are C1 hypersurfaces:

Theorem 3.5 (Higher regularity for F -minimizers).Assume that c0 is symmetric, that c0 and c1 satisfy (A) and are Lipschitz con-

tinuous in space. Let F ∈ P, and let E be a minimizer of F(h, k, ·, F ) on P. Setg(p) = ±d∂F (p), where we take the − sign if p ∈ F , and the + sign otherwise.

Let p ∈ ∂E be such that ∂E is a C1 hypersurface near p: there exist R > 0,M > 0 and a C1 function f : BN−1

R (p) → (−M,M) such that, possibly rotating andrelabelling, we have

E ∩ (BN−1R (p) × (−M,M)) = (x, y); x ∈ BN−1

R (p),−M < y < f(x).

Then f is of class C2,α in BN−1R (p) for some 0 < α < 1 and satisfies

1

hg((x, f(x))) = ∆f(x) + c0(·, kh) ⋆ 1E((x, f(x))) + c1((x, f(x)), kh). (3.4)

112

4. The upper and lower limits

Therefore the mean curvature Hp of ∂E at p verifies

1

hg(p) = Hp + c0(·, kh) ⋆ 1E(p) + c1(p, kh). (3.5)

Proof. We begin by proving that f satisfies (3.4) in the sense of distributions. This issimply the Euler-Lagrange equation for F , and the proof is the same as that of Am-brosio (see [6], after the statement of Theorem 3.3), with the additional observationthat the first variation of

K 7→ 1

2

∫

K

c0(·, kh) ⋆ 1K(x) dx, K 7→∫

K

c1(x, kh) dx

in the direction of a C2 vector field Φ is respectively

K 7→∫

∂K

c0(·, kh) ⋆ 1K(x) 〈Φ(x), νx〉 dHN−1(x)

and K 7→∫

∂K

c1(x, kh) 〈Φ(x), νx〉 dHN−1(x),

where νx is the outer unit normal to K at x ∈ ∂K. The symmetry of c0 is usedhere, along with the continuity of c1 and c0 ⋆ 1K in space.

Knowing this, we apply Theorem 1.2 p. 219 of [45] to f and to each of the ∂f∂xi

, toconclude that f is C2,α in BN−1

R (p). This last assertion uses the Lipschitz continuityof c1 and c0 ⋆ 1K in space. Both conclusions immediately follow.

4 The upper and lower limits

In this section we are going to prepare the proofs of Theorems 1.4 and 1.5. LetE be a minimizing movement with initial condition E0 and let (hn) be a sequencesuch that Ehn([t/hn]) converges to E(t) in L1(RN ) for all t ∈ [0, T ] as n goes toinfinity. We define the upper and lower limits of the sets Ehn(k) for n → +∞ andk ∈ N as follows:

E∗(t) = x ∈ RN ; ∃(hn′) ⊂ (hn), kn′ → +∞ and xn′ ∈ Ehn′ (kn′)

with kn′hn′ → t and xn′ → x,E∗(t) = R

N \ x ∈ RN ; ∃(hn′) ⊂ (hn), kn′ → +∞ and xn′ /∈ Ehn′ (kn′)

with kn′hn′ → t and xn′ → x.By construction, E∗ is closed while E∗ is open, and E∗(t) ⊂ E(t) ⊂ E∗(t) for

all t ∈ [0, T ] and almost everywhere in RN . Indeed E∗(t) and E∗(t) are definedrespectively as the sets of cluster points of sets Ehn(k) and RN \ Ehn(k) for allk → +∞ such that khn → t, and, passing to a subsequence and removing a set ofzero LN measure, our minimizing movement at time t, E(t), was constructed as thepointwise limit of sets Ehn(k) for some such k = [t/hn].

We will use the regularity result Theorem 3.5 to estimate the normal velocityof the evolutions t 7→ E∗(t) and t 7→ E∗(t) in function of E. Then we will prove aregularity result for E∗ and E∗, and compare the initial sets E∗(0), E∗(0) and E0.

113


In order that our minimizing procedure be consistent with the evolution law (1.1)as ensured by Theorem 3.5, we will assume in particular throughout this section thatc0 is symmetric.

4.1 Velocity of E∗ and E∗

Here we are going to prove a rigorous version of the heuristic fact that E∗ moveswith velocity

Vx,t ≤ Hx,t + c0(·, t) ⋆ 1E(t)(x) + c1(x, t),

while E∗ moves with velocity

Vx,t ≥ Hx,t + c0(·, t) ⋆ 1E(t)(x) + c1(x, t),

where Hx,t respectively denotes the mean curvature of ∂E∗ and ∂E∗. FollowingCardaliaguet [30], we formulate this statement in terms of test functions: let us firstdefine the classical mean curvature operator

h(p,X) = Trace(X) − 〈Xp, p〉|p|2 ,

for X ∈ SN and p ∈ RN \0, and let us define, for any subset A of RN , A = RN \ A,and for any subset B of RN × [0, T ], B = (RN × [0, T ]) \B.

Proposition 4.1. Under the assumptions of Theorem 1.4, we have:

1. For any t ∈ (0, T ), if a test function φ of class C2 has a local maximum on E∗

at some point (x, t) ∈ ∂E∗, with Dφ(x, t) 6= 0, then

φt(x, t) ≥ h(Dφ(x, t), D2φ(x, t)) −[c0(·, t) ⋆ 1E(t)(x) + c1(x, t)

]|Dφ(x, t)|.

2. For any t ∈ (0, T ), if a test function φ of class C2 has a local minimum on E∗ atsome point (x, t) ∈ ∂E∗, with Dφ(x, t) 6= 0, then

φt(x, t) ≤ h(Dφ(x, t), D2φ(x, t)) −[c0(·, t) ⋆ 1E(t)(x) + c1(x, t)

]|Dφ(x, t)|.

Proof. We only prove the first point, the proof of the second being similar. Lett ∈ (0, T ) and φ of class C2 have a local maximum on E∗ at some point (x, t) ∈ ∂E∗,with Dφ(x, t) 6= 0. We can assume without loss of generality that it is a strictmaximum. By definition of E∗, there exist kn → +∞ and xn ∈ ∂Ehn(kn) withknhn → t and xn → x, such that φ has a local maximum (that we can assume to bestrict) on Ehn = ∪k Ehn(k)×khn at (xn, knhn), with Dφ(xn, knhn) 6= 0. It followsthat Γhn(kn) = x ∈ RN ;φ(x, knhn) = φ(xn, knhn) is a smooth exterior contactsurface to Ehn(kn) at xn, and therefore Theorems 3.4 and 3.5 imply that ∂Ehn(kn)is a C2,α hypersurface near xn. We now infer from the local relative position of Γand ∂Ehn(kn) that the curvature of ∂Ehn(kn) at xn, Hn

xn, is less than the curvature

of Γ at xn:

Hnxn

≤ − 1

|Dφ(xn, knhn)|h(Dφ(xn, knhn), D2φ(xn, knhn)).

114


Now if kn ≥ 1, (3.5) implies that

± 1

hnd∂Ehn(kn−1)(xn) = Hn

xn+ c0(·, knhn) ⋆ 1Ehn (kn)(xn) + c1(xn, knhn),

where we take the − sign if xn ∈ Ehn(kn − 1), and the + sign otherwise. With thisconvention,

± 1

hnd∂Ehn(kn−1)(xn) ≥ ± 1

hndΓhn (kn−1)(xn) = − φt(xn, knhn)

|Dφ(xn, knhn)| + o(1).

Putting together the last three equations yields

φt(xn, knhn) + o(1) ≥ h(Dφ(xn, knhn), D2φ(xn, knhn))

−[c0(·, knhn) ⋆ 1Ehn(kn)(xn) + c1(xn, knhn)

]|Dφ(xn, knhn)|. (4.1)

Thanks to the discrete Hölder estimate, Theorem 2.4, we know, since knhn → t,that Ehn(kn) → E(t) in L1(RN). Passing to a subsequence, we can assume thatEhn(kn) → E(t) almost everywhere. As a consequence, sending n to +∞, we getthe result, namely:

φt(x, t) ≥ h(Dφ(x, t), D2φ(x, t)) −[c0(·, t) ⋆ 1E(t)(x) + c1(x, t)

]|Dφ(x, t)|.

4.2 Regularity of E∗ and E∗

Now we are going to prove a regularity result for the tubes E∗ and E∗ whichallows in particular to treat the degenerate case Dφ(x, t) = 0 in Proposition 4.1:

Proposition 4.2. For all x in RN , the maps t 7→ dE∗(t)(x) and t 7→ dcE∗(t)(x) areleft-continuous on (0, T ].

To prove this we first need to estimate in a finer way than what we have donein Section 2 how Eh(k) can expand or shrink at most at each iteration. This isthe equivalent of [1, Theorem 5.4]. Let us first define for simplicity of forthcomingestimates the scaled ball WR = BR/(ωN )1/N (0) = BR/ω∗(0), so that LN(WR) = RN .Then WR minimizes the perimeter among all sets E ∈ P such that LN(E) = RN .This property will provide the necessary estimates.

Let us also define, for any subsets A and B of RN , A−B = R

N \ ((RN \A)+B).

Lemma 4.3. Let F ∈ P and let E be a minimizer of F(h, k, ·, F ) on P. Let L bedefined as in (1.5), and R(h) = 2Lω∗h+ 2

√L2ω2

∗h2 + 2ω∗hP (W1). Then

F −WR(h) ⊂ E ⊂ F +WR(h) a.e.

115


Proof. We begin by proving the left-hand side inclusion, and we will see that theother inclusion immediately follows. We adapt the proofs of [1, Section 5].

Step 1. Let us first prove that if 0 < R < S, WS ⊂ F and 0 < 2LN(WR \E) ≤ RN ,then

S − R

ω∗hR− 2LR ≤ N − 1

NP (W1) +

21/N(N − 1)

N2P (W1)

LN(WR \ E)

RN.

We compare E and E ∪WR with respect to the functional F(h, k, ·, F ):

P (E) +1

h

∫

E∆F

d∂F (x) dx−∫

E

(1

2c0(·, kh) ⋆ 1E(x) + c1(x, kh)

)dx

≤ P (E ∪WR) +1

h

∫

(E∪WR)∆F

d∂F (x) dx

−∫

E∪WR

(1

2c0(·, kh) ⋆ 1E∪WR

(x) + c1(x, kh)

)dx.

Since WR ⊂ F , we check that E∆F = ((E ∪WR)∆F ) ∪ (WR \ E). This, togetherwith manipulations similar to those of previous proofs, implies that

P (E ∪WR) − P (E) ≥ 1

h

∫

WR\Ed∂F (x) dx− 2LLN(WR \ E)

≥(S −R

ω∗h− 2L

)LN(WR \ E),

(4.2)

since the inclusion WS ⊂ F implies that d∂F (x) ≥ (S−R)/ω∗ for each x ∈WR. Butconclusion (4) of [1, Proposition 5] implies that

P (E ∪WR) − P (E)

≤RN−1P (W1)

N − 1

N

LN(WR \ E)

RN+

21/N(N − 1)

N2

(LN(WR \ E)

RN

)2,

and the result follows from the last two inequalities.

Step 2. Now let us assume that the conclusion of the lemma does not hold, i.e. thatif we set A = (F −WR(h)) \ E, then LN(A) > 0. There must exist p ∈ A such thatfor any r > 0, LN(A∩Br(p)) > 0. We can assume, possibly applying a translation,that p = 0. Therefore WR(h) ⊂ F and LN(WR(h)/2 \ E) > 0. Moreover we also have

2LN(WR(h)/2 \ E) ≤(R(h)

2

)N

,

otherwise we would obtain as in Step 1 with S = R(h) and R = R(h)/2 that

P (E ∪WR(h)/2) − P (E) ≥(R(h)

2ω∗h− 2L

)LN(WR(h)/2 \ E)

>

(R(h)

2ω∗h− 2L

)1

2

(R(h)

2

)N

,

116


because R(h)2ω∗h

− 2L > 0. But P (E ∪WR(h)/2) ≤ P (E) + P (WR(h)/2), whence

(R(h)

2ω∗h− 2L

)1

2

(R(h)

2

)N

< P (WR(h)/2) =

(R(h)

2

)N−1

P (W1),

or equivalently1

ω∗h

(R(h)

2

)2

− LR(h) < 2P (W1),

which is contradictory with the choice of R(h), since equality should hold instead ofthe last inequality. Then we can apply Step 1 with S = R(h) and R = R(h)/2 toinfer that

1

ω∗h

(R(h)

2

)2

− LR(h) ≤ N − 1

NP (W1) +

21/N (N − 1)

N2P (W1)

LN(WR(h)/2 \ E)

(R(h)/2)N,

or thanks to the choice of R(h):

2 ≤ N − 1

N+

21/N (N − 1)

N2

1

2,

which is false. This proves the left-hand side inclusion of Lemma 4.3.

Step 3. Let us now explain why the left-hand side inclusion is sufficient to deducethe right-hand side one. Let B = BD(0) be a large ball. It is easy to check thatif F ∈ P with F ⊂ BD−1(0), and if E ∈ P with E ⊂ BD−1(0) is a minimizer ofF(h, k, ·, F ) on P, then B \ E is a minimizer of

E 7→ P (E) +1

h

∫

E∆(B\F )

d∂F (x) dx−∫

E

(1

2c0(·, kh) ⋆ 1E(x) + c1(x, kh)

)dx

among all sets in P and included in B, where c1(x, kh) = −c1(x, kh) + c0(·, kh) ⋆1B(x). Therefore, taking h small enough so that R < 1, the arguments on E andF in the previous steps transform into the same arguments for B \ E and B \ F ,since in particular the term 2L appearing in (4.2) was taken so large (with the apriori useless factor 2) as to get the lower bound there also with c1 in place of c1.The conclusion F −WR ⊂ E transforms into (B \F )−WR ⊂ B \E, that is exactlyE ⊂ F +WR.

The last lemma provides a bound on the growth of F -minimizers at each iterationequal to 2Lω∗h+ 2

√L2ω2

∗h2 + 2ω∗hP (W1), and of the order of

√h. This is not fine

enough to conclude the left continuity, mainly because if kh→ t, then k√h→ +∞

and the bound is lost in the limit movement. The following lemma refines the boundto the order h.

Lemma 4.4. Set δ = 2N−1NP (W1) and R(h) = 2Lω∗h+ 2

√L2ω2

∗h2 + 2ω∗hP (W1).

1. Assume that p+WS ⊂ Eh(k) a.e. for some p ∈ RN and k, h such that kh ≤ T .If h and j are small enough so that R(h) < S

4and jh ≤ min S2

4ω∗(δ+2LS), T − kh,

thenp+WS−ω∗(

δS

+2L)jh ⊂ Eh(k + j) a.e.

117


2. Assume that p +WS ⊂ RN \ Eh(k) a.e. for p ∈ R

N and k, h such that kh ≤ T .If h and j are small enough so that R(h) < S

4and jh ≤ min S2

4ω∗(δ+2LS), T − kh,

thenp+WS−ω∗(

δS

+2L)jh ⊂ RN \ Eh(k + j) a.e.

Proof. Let us prove the first assertion. For simplicity we assume without loss ofgenerality that p = 0. We prove the result by induction on j. The result forj = 0 is the assumption. Let us assume that the result holds for some j such that(j + 1)h ≤ min S2

4ω∗(δ+2LS), T − kh. We know thanks to Lemma 4.3 that

Eh(k + j) −WR(h) ⊂ Eh(k + j + 1) a.e. (4.3)

Since the induction assumption states that WS−ω∗( δS

+2L)jh ⊂ Eh(k + j), and sincethe assumptions on j and h imply that

R(h) <S

2− ω∗

(δ

S+ 2L

)jh,

we deduce from (4.3) that WS/2 ⊂ Eh(k + j + 1) almost everywhere. Let us set

rmax = supr;Wr ⊂ Eh(k + j + 1) a.e. ≥ S

2.

Step 1 of Lemma 4.3 shows, by sending R to r+max, that

1

ω∗h

(S − ω∗

(δ

S+ 2L

)jh

− rmax

)rmax − 2Lrmax ≤ N − 1

NP (W1) =

δ

2,

from which we infer thatS − ω∗

(δ

S+ 2L

)jh

− rmax ≤

(δ

2rmax+ 2L

)ω∗h ≤ ω∗

(δ

S+ 2L

)h,

and the result for Eh(k + j + 1) follows, so that the proof by induction is complete.The proof of the second point is entirely identical, according to the remark in Step3 of the proof of Lemma 4.3.

We are now ready to prove Proposition 4.2. This proof is inspired by the proofof Lemma 4.7 of [31].

Proof of Proposition 4.2. Let us start with E∗. Assume on the contrary of our claimthat there exist x ∈ RN and t ∈ (0, T ] such that s 7→ dE∗(s)(x) is not left continuousat t. Since this map is lower semi-continuous thanks to the closedness of E∗, wededuce that there exist ε > 0 and a sequence (tp) converging to t− such that for allp ∈ N,

dE∗(tp)(x) > dE∗(t)(x) + ε.

Let S = εω∗, so that WS is the closed ball of radius ε centered at 0. By consideringa projection of x on E∗(t), we can assume that x ∈ E∗(t) and for all p ∈ N,

dE∗(tp)(x) > ε.

118


Set for a fixed p, kn = [tp/hn], so that knhn → t−p as n → +∞. By definitionof E∗(tp), there exists n0 large enough depending on p so that for all n ≥ n0,dEhn (kn)(x) > ε. Let us set

M =

(δ

S+ 2L

).

Then we can apply assertion 2 of Lemma 4.4 to deduce that for all n ≥ n0 such thatR(hn) < S

4and for all j such that jhn ≤ min S2

4ω∗(δ+2LS), T − knhn,

dEhn(kn+j)(x) ≥ ε−Mjhn. (4.4)

Indeed we have WS−ω∗(δS

+2L)jhn(x) ⊂ RN \ Ehn(kn + j). Let us set

τ = min

ε

2M,

S2

4ω∗(δ + 2LS)

and fix s ∈ (0, τ) with s ≤ T−tp. We set jn = [s/hn] so that jnhn → s− as n→ +∞.Then

jnhn ≤ min

S2

4ω∗(δ + 2LS), T − knhn

for n large enough, so that sending n to +∞ in (4.4) yields, by definition of E∗(tp+s),

dE∗(tp+s)(x) ≥ ε−Ms ≥ ε

2.

Taking s = t − tp for p big enough so that 0 < s < τ , we get dE∗(t)(x) ≥ ε2, which

contradicts the fact that x ∈ E∗(t). The proof for dcE∗is obtained in the same way

by using assertion 1 of Lemma 4.4.

4.3 Comparison at initial time

We finish by giving a consequence of the previous growth results on the com-parison of the initial sets E∗(0) and E∗(0) with E0. This result will be essential forcomparison at later times:

Proposition 4.5. We haveE0⊂ E∗(0) ⊂ E∗(0) ⊂ E0.

Proof. We only prove that E∗(0) ⊂ E0, the left-hand side inclusion is obtained bysimilar arguments. Suppose on the contrary that there exists x ∈ E∗(0) \E0. Thenwe can find some ε > 0 such that Bε(x) ⊂ R

N \ E0. By definition of E∗(0), thereexist sequences (kn) converging to +∞ and (xn) converging to x with knhn → 0 andxn ∈ Ehn(kn). Thanks to Lemma 4.4 and the facts that Ehn(0) = E0 and knhn → 0,we know that there exists M > 0 depending only on ε, L and N such that if n islarge enough, then

Bε−Mknhn(x) ⊂ RN \ Ehn(kn).

But xn → x and ε −Mknhn → ε, so that xn ∈ Bε−Mknhn(x) for n large enough.This is a contradiction since xn ∈ Ehn(kn), and this proves the proposition.

119


5 Minimizing movements and weak solutions

With the tools of Section 4, we are now ready to prove Theorem 1.4. Since weknow that E∗(t) ⊂ E(t) ⊂ E∗(t) a.e. for all t ∈ [0, T ], it suffices to prove that forall t ∈ [0, T ],

u(·, t) > 0 ⊂ E∗(t) and E∗(t) ⊂ u(·, t) ≥ 0.

To this end, we will use a comparison principle for discontinuous viscosity solutions.We therefore start by giving equations satisfied by 1E∗ and 1E∗ in the viscosity sense,in relation with Theorem 4.1:

Theorem 5.1. Under the assumptions of Theorem 1.4, we have:

1. For any (x, t) ∈ RN × (0, T ), if a test function φ of class C2 is such that 1E∗ − φhas a local maximum at (x, t), then:

• if Dφ(x, t) 6= 0, we have

φt(x, t) ≤ h(Dφ(x, t), D2φ(x, t)) +[c0(·, t) ⋆ 1E(t)(x) + c1(x, t)

]|Dφ(x, t)|.

• if Dφ(x, t) = 0 and D2φ(x, t) = 0, we have

φt(x, t) ≤ 0.

2. For any (x, t) ∈ RN × (0, T ), if a test function φ of class C2 is such that 1E∗ − φ

has a local minimum at (x, t), then:

• if Dφ(x, t) 6= 0, we have

φt(x, t) ≥ h(Dφ(x, t), D2φ(x, t)) +[c0(·, t) ⋆ 1E(t)(x) + c1(x, t)

]|Dφ(x, t)|.

• if Dφ(x, t) = 0 and D2φ(x, t) = 0, we have

φt(x, t) ≥ 0.

Proof. We only prove the first point, since the second point uses the same arguments.We only need to consider the case where (x, t) ∈ ∂E∗, since otherwise all derivativesof φ at (x, t) vanish and the equation is obviously satisfied.

First case: Dφ(x, t) 6= 0. In this case it is straightforward to check that −φ hasa local maximum on E∗ at (x, t). Therefore, the first point of Proposition 4.1 givesthe result.

Second case: Dφ(x, t) = 0 and D2φ(x, t) = 0. We can always assume that ourmaximum is equal to 0, i.e. φ(x, t) = 1E∗(x, t) = 1. Let us also assume thatφt(x, t) > 0. Then a Taylor expansion of φ at (x, t) shows that there exist δ > 0and k > 0 such that for all (y, s) satisfying s ∈ (t− δ, t) and |y − x| < 2k(t− s)1/3,then 1E∗(y, s) ≤ φ(y, s) < φ(x, t) = 1, whence y /∈ E∗(s). As a consequence for alls ∈ (t− δ, t),

dE∗(s)(x) > k(t− s)1/3.

120

6. Comparison with the smooth flow

Now we can proceed as in the proof of Proposition 4.2, using the growth controlgiven by Lemma 4.4, to prove that there are positive constants k1 and k2 such thatfor all s < t close enough to t,

dE∗(t)(x) > k(t− s)1/3 −(

k1

(t− s)1/3+ k2

)(t− s) > 0,

which contradicts the fact that x ∈ E∗(t). Therefore φt(x, t) ≤ 0.

Proof of Theorem 1.4. The previous theorem shows that 1E∗ is a subsolution of thelevel-set equation (1.4), while 1E∗ is a supersolution. Indeed, an argument of Barlesand Georgelin [19, Proposition 1] shows that under the conclusions of Theorem5.1 there is no property to check when the test function satisfies Dφ(x, t) = 0and D2φ(x, t) 6= 0. To conclude we use a method initiated by Barles, Soner andSouganidis [22, Theorem 2.1]: let (Φn) be a sequence of smooth functions such thatΦn ≡ 1 on [0,+∞), Φ′

n ≥ 0 in R, Φn(R) ⊂ [0, 1] and infn Φn = 0 on (−∞, 0).Thanks to Proposition 4.5, we know that 1E∗(0) ≤ Φn(u0) in RN . Since (1.4) is ageometric equation, Φn(u) is a uniformly continuous solution of this equation. Thecomparison principle [22, Theorem 1.3] implies that for all t ∈ [0, T ),

1E∗(t) ≤ Φn(u(·, t)).If x ∈ u(·, t) < 0, we therefore have

1E∗(t)(x) ≤ infn

Φn(u(x, t)) = 0,

which means that x /∈ E∗(t). As a consequence E∗(t) ⊂ u(·, t) ≥ 0 for allt ∈ [0, T ), which also holds for t = T by continuity of u and thanks to Proposition4.2. The argument to prove that u(·, t) > 0 ⊂ E∗(t) is similar.

In case there is no fattening, we deduce that for all t ∈ [0, T ], E(t) = u(·, t) ≥ 0almost everywhere, and we can replace E(t) by u(·, t) ≥ 0 in (1.4) to deduce thatu is a viscosity solution of (1.4). This concludes the proof of Theorem 1.4.

6 Comparison with the smooth flow

Now we are going to show that our construction is consistent with smooth flowsif they exist: we turn to the proof of Theorem 1.5. Following Cardaliaguet andPasquignon [32], we define a sub/super pair of solutions for our nonlocal motion.Roughly speaking, it is a pair (K1,K2) of tubes, where K1 moves with velocity

Vx,t ≤ Hx,t + infK1(t)⊂K⊂K2(t)

c0(·, t) ⋆ 1K(x) + c1(x, t),

while K2 moves with velocity

Vx,t ≥ Hx,t + supK1(t)⊂K⊂K2(t)

c0(·, t) ⋆ 1K(x) + c1(x, t).

As we did at the beginning of Section 4.1, we formulate this in terms of testfunctions:

121


Definition 6.1 ([32], Definition 2.5). Let K1 and K2 be compact subsets of RN such

that K1 ⊂K2. A sub/super pair of solutions with initial data (K1, K2) is a pair

(K1,K2) of tubes such that1. K1 ⊂ K2.2. K1(0) = K1 and K2(0) ⊂ K2.3. For any t ∈ (0, T ), if a test function φ of class C2 has a local maximum on K1

at some point (x, t) ∈ ∂K1, then

φt(x, t) ≥ h(Dφ(x, t), D2φ(x, t))−[

infK1(t)⊂K⊂K2(t)

c0(·, t) ⋆ 1K(x) + c1(x, t)

]|Dφ(x, t)|.

4. For any t ∈ (0, T ), if a test function φ of class C2 has a local minimum on K2 atsome point (x, t) ∈ ∂K2, then

φt(x, t) ≤ h(Dφ(x, t), D2φ(x, t))−[

supK1(t)⊂K⊂K2(t)

c0(·, t) ⋆ 1K(x) + c1(x, t)

]|Dφ(x, t)|.

Such sub/super pairs of solutions exist and we can define, following Cardaliaguetand Pasquignon, extremal sub/super pairs of solutions (Kε

1,Kε2) with initial data

(E0 − εB1(0), E0 + εB1(0)). The extremality holds with respect to the inclusion.Moreover, if E0 is compact with uniformly C3+α boundary, and if Er is a smoothevolution with C3+α boundary, starting from E0 with normal velocity given by (1.9),then Kε

1 ⊂ Er ⊂ Kε2 and both Kε

1 and Kε2 converge to Er in the Hausdorff distance

as ε → 0, as proved by Cardaliaguet [30]. This implies in particular that a smoothevolution with C3+α boundary is necessarily unique.

Now, owing to the respective velocities of Kε1, E∗, E∗ and Kε

2, we want to comparethese sets. Going through the corresponding proofs in [32] and [30], we check thatthe estimation on the velocities of E∗ and E∗ (Proposition 4.1), their regularityproperty (Proposition 4.2) and their initial position relatively to E0 (Proposition4.5) give the following result:

Theorem 6.2 ([32], Theorem 2.11). Under the assumptions of Theorem 1.5, let(Kε

1,Kε2) be an extremal sub/super pair of solutions with (E0 − εB1(0), E0 + εB1(0))

as initial data. If Kε1(t) and Kε

2(t) are non-empty for all t ∈ [0, T ], then

Kε1(t) ⊂ E∗(t) ⊂ E∗(t) ⊂ Kε

2(t) for all t ∈ [0, T ).

We are finally ready to prove Theorem 1.5.

Proof of Theorem 1.5. Since Kε1 and Kε

2 converge to the smooth evolution Er start-ing from E0 in the Hausdorff distance if the latter exists, we deduce that for allt ∈ [0, T ), E∗(t) = E∗(t) = Er(t). This also holds for t = T thanks to Proposition4.2. Moreover we know that for all t ∈ [0, T ], E∗(t) ⊂ E(t) ⊂ E∗(t) a.e., so theresult follows.

122

7. Existence and uniqueness of a smooth solution

7 Existence and uniqueness of a smooth solution

To conclude this work, it is natural to verify that such a smooth evolution exists(we already know that it must be unique). This is the claim of Theorem 1.6, thatwe prove now, using a fixed point method. We therefore begin by constructing asmooth solution for the local problem (i.e. with prescribed velocity).

7.1 Existence of smooth solutions for the local problem

Theorem 7.1 (Existence of a smooth solution for the local problem).Assume that E0 is a compact subset of RN with uniformly C3+α boundary and

that c ∈ W 2,1;∞(RN × [0, T ]). Then there exist a small time t0 > 0 depending onlyon E0 and on an upper bound on ‖c‖W 2,1;∞(RN×[0,T ]), and a smooth evolution Er withC3+α boundary defined on [0, t0], starting from E0, with normal velocity

Vx,t = Hx,t + c(x, t), (7.1)

where Hx,t is the mean curvature of Γ(t) = ∂Er(t) at x.

The proof is an adaptation of the one proposed by Evans and Spruck [39] forthe classical mean curvature motion (see also Giga, Goto [47] and Maekawa [57] formore general equations). For the reader’s convenience, we give the steps of the proofto explain how to treat the dependence in the space variable of the velocity c.

Assume we are given the smooth hypersurface Γ0 = ∂E0, a time t0 > 0 and asmooth evolution t 7→ Γ(t) = ∂E(t) of surfaces developing from Γ0 on [0, t0] withnormal velocity Vx,t. Heuristically, one can show (see [39]) that the signed distancefunction d to Γ(t) defined by

d(x, t) =

−dist(x,Γ(t)) if x ∈ RN \ E(t)dist(x,Γ(t)) if x ∈ E(t)

is a solution ofvt = F (D2v, v) + c(x− v(x, t)Dv(x, t), t) (7.2)

with

F (R, z) = f(λ1(R), ..., λn(R), z) =N∑

i=1

λi(R)

1 − λi(R)z,

where λ1(R) ≤ λ2(R) ≤ ... ≤ λN(R) are the eigenvalues of R. F is a priori definedand smooth for |R| and |z| small enough, but we extend it to be smooth on all ofSN × R with |F |, |DF | and |D2F | bounded as in [39].

The idea is to study directly the PDE (7.2). To this end, we set Γ0 = ∂E0 andlet

g(x) =

−dist(x,Γ0) if x ∈ RN\E0

dist(x,Γ0) if x ∈ E0(7.3)

be the signed distance function to Γ0. We fix δ0 so small that g is of class C3+α

withinV = x ∈ R

N , −δ0 < g(x) < δ0

123


and we set, for t0 > 0 to be determined,

Q = V × (0, t0), Σ = ∂V × [0, t0].

The plan is to consider a solution to the PDE

vt = F (D2v, v) + c(x− vDv, t) in Q|Dv|2 = 1 on Σv = g on V × t = 0

(7.4)

and prove that the zero level sets of v(·, t) are smooth hypersurfaces evolving withnormal velocity given by (7.1).

First, we have the following existence result for this nonlinear PDE (see the bookby Lunardi [56, Theorem 8.5.4 and Proposition 8.5.6]):

Theorem 7.2 (Existence for the non-linear PDE).There exist δ0 depending only on E0 and t0 > 0 depending only on E0 and

on an upper bound on ‖c‖W 2,1;∞(RN×[0,T ]) such that there exists a unique solutionv ∈ C2+α, 2+α

2 (Q) of the PDE (7.4). Moreover the first order space derivatives vxk,

for 1 ≤ k ≤ N , belong to C2+α, 2+α2 (V × [0, t0]).

Evolution of the zero level set of v

The rest of the proof is devoted to showing that, possibly reducing t0, the map-ping

t ∈ [0, t0] 7→ Er(t) = (E0 \ V ) ∪ x ∈ V, v(x, t) ≥ 0is a smooth evolution with C3+α boundary, with normal velocity given by (7.1).

Proposition 7.3 (Distance property of v).Let v be the solution of (7.4) given by Theorem 7.2. Then we have

|Dv|2 = 1 in Q. (7.5)

Proof. We adapt the proof of Evans and Spruck [39, Theorem 3.1].

Step 1. Let w = |Dv|2 − 1. Then w ∈ C2+α, 2+α2 (V × [0, t0]). Moreover, using the

PDE (7.4) and the definition of g given by (7.3), we get that

w = 0 on Σ ∪ (V × t = 0).Step 2. Differentiating (7.4), we compute (with implicit summations over i, j, k)

vtxk=

∂F

∂rij(D2v, v)vxixjxk

+∂F

∂z(D2v, v)vxk

+∂

∂xkc(x− vDv, t).

Therefore

wt =2vxkvxkt

=2∂F

∂rij(D2v, v)vxk

vxkxixj+ 2

∂F

∂z(D2v, v)|Dv|2 + 2

∂

∂xk(c(x− vDv, t))vxk

=∂F

∂rij(D2v, v)wxixj

− 2∂F

∂rij(D2v, v)vxkxi

vxkxj

+ 2∂F

∂z(D2v, v)|Dv|2 + 2

∂


. (7.6)

124


Now we compute

2∂


=2∂c

∂xi(x− vDv, t)(δik − vxk

vxi− vvxkxi

) vxk

= − 2 (|Dv|2 − 1)∂c

∂xi(x− vDv, t) vxi

− ∂c

∂xi(x− vDv, t) v wxi

= − w l1(x, t) − wxil2,i(x, t),

where

l1(t, x) = 2∂c

∂xi(x− vDv, t) vxi

and

l2,i(x, t) =∂c

∂xi

(x− vDv) v.

Moreover as recalled in [39],

∂F

∂rij

(D2v)vxkxivxkxj

=∂F

∂z(D2v, v).

As a consequence (7.6) becomes

wt =∂F

∂rij

(D2v, v)wxixj+

(2∂F

∂z(D2v, v) − l1(x, t)

)w − l2,i(x, t)wxi

.

In view of the uniform ellipticity of F (see [39, Lemma 2.1]), we see that this is auniformly parabolic equation. Using the fact that w = 0 on the parabolic boundaryof Q, we deduce that w = 0 in Q. This ends the proof of the proposition.

Now, using (7.5), we get that

Γ = (x, t) ∈ Q, v = 0

is a C1 hypersurface in Q and each slice Γ(t) = x ∈ V, v(x, t) = 0 is a C3+α

hypersurface in V . Moreover we have the following equivalent of [39, Theorem 3.2]:

Theorem 7.4 (Existence of a classical evolution).The surfaces Γ(t)0≤t≤t0 comprise a classical motion starting from Γ0 with nor-

mal velocityVx,t = Hx,t + c(x, t).

Given that Γ(t) = ∂Er(t) for all t ∈ [0, t0], provided t0 is small enough dependingonly on δ0 and an upper bound on ‖c‖W 2,1;∞(RN×[0,T ]), this concludes the proof ofTheorem 7.1.

125


7.2 Existence of a smooth solution for the nonlocal problem

With the results of the previous section, we are now ready to carry out the fixedpoint argument. We use the same notation as in the previous section, in particularF , Q, Σ and V , with the same δ0 fixed, but for some t0 to be determined. Usingthe same method as in Section 7.1, our goal is to construct a solution to the PDE

vt = F (D2v, v) + (c0(·, t) ⋆V 1v(·,t)≥0)(x− vDv, t) + c(x− vDv, t) in Q

|Dv|2 = 1 on Σ

v = g on V × t = 0(7.7)

where ⋆V denotes the convolution restricted to V , i.e.

c0(·, t) ⋆V 1v(·,t)≥0(x) =

∫

V

c0(x− y, t)1v(·,t)≥0(y) dy,

and

c(x, t) =

∫

E0\Vc0(x− y, t)dy + c1(x, t).

We define the set

E =

v ∈ C2+α, 2+α

2 (Q)

∣∣∣∣∣∣∣∣

‖v − g‖C2+α,2+α

2 (Q)≤ R0

|Dv|2 = 1 in Qv = g on V × t = 0vt = h0 on V × t = 0

,

where g is defined by (7.3), R0 is a small constant which will be precised later and

h0 = F (D2g, g) + c0 ⋆ 1E0(x− gDg, 0) + c1(x− gDg, 0).

For w ∈ E, we set

cw(x, t) = c0(·, t) ⋆V 1w(·,t)≥0(x) + c(x, t).

Under the assumptions on c0 and c1 it is easy to check that cw ∈W 2,1;∞(RN × [0, T ])(see the definition of W 2,1;∞(RN × [0, T ]) after (1.11)). Indeed, the only difficulty isto check that cw is Lipschitz in time. To do this, let us state the following lemma:

Lemma 7.5 (Estimate on characteristic functions).There exists a constant C such that if u1, u2 ∈ C1(V ) satisfy 〈Dui, Dg〉 ≥ 1

2in

V for i = 1, 2, then

‖1u1≥0 − 1u2≥0‖L1(V ) ≤ C‖u1 − u2‖L∞(V ).

The proof is an easy adaptation of [3, Lemma 42] (using local cards and apartition of unity), so we skip it.

For any u ∈ E, Du satisfies Du(·, 0) = Dg and is Hölder in time. As a conse-quence, for t0 small enough depending only on an upper bound on

‖u‖C2+α,2+α

2 (Q)≤ R0 + ‖g‖C2+α(V ),

126


we have 〈Du(·, t), Dg〉 ≥ 1/2 in V for any u ∈ E and t ∈ [0, t0]. Therefore, usingthe previous lemma, we can compute

|cw(x, t) − cw(x, s)|= |c0(·, t) ⋆V 1w(·,t)≥0(x) − c0(·, s) ⋆V 1w(·,s)≥0(x) + c(x, t) − c(x, s)|≤ |c0(·, t) ⋆V 1w(·,t)≥0(x) − c0(·, t) ⋆V 1w(·,s)≥0(x)|

+ |c0(·, t) ⋆V 1w(·,s)≥0(x) − c0(·, s) ⋆V 1w(·,s)≥0(x)| + |c(x, t) − c(x, s)|≤Cw|t− s|,

whereCw =C‖c0‖L∞(RN×[0,T ])‖w‖C2+α,2+α

2 (Q)

+2‖c0‖W 1,∞([0,T ];L∞(RN ))LN(E0) + ‖c1‖W 2,1;∞(RN×(0,T )).

The factor 2 appears if we assume that LN(V \ E0) ≤ LN(E0), which is alwayspossible. We remark that this constant Cw can be chosen independently of w sincewe have

‖w‖C2+α,2+α

2 (Q)≤ R0 + ‖g‖C2+α(V ).

This, together with similar estimates on space derivatives, implies that for anyw ∈ E,

‖cw‖W 2,1;∞(Q) ≤ C(1 +R0),

where the constant C does not depend on t0, R0.

As a consequence of Theorem 7.2, for t0 small enough (depending only on R0),we can therefore define for any w ∈ E, v = Φ(w) as the unique solution of

vt = F (D2v, v) + cw(x− vDv, t) in Q|Dv|2 = 1 on Σv = g on V × t = 0.

Moreover the proof of Theorem 7.2 shows that provided t0 is small enough (depend-ing only on R0), then v ∈ E for any w ∈ E. Let us now prove that Φ is a contraction,for a good choice of parameters R0 and t0.

Let w1, w2 ∈ E, v1 = Φ(w1), v2 = Φ(w2) and v = v2 − v1. Then v is a solutionof

vt − aijvxixj+ fivxi

+ ev = δ + A(D2v,Dv, v, x, t) in Q∂v

∂ν= a(Dv, x, t) on Σ

v = 0 on V × t = 0,

where

aij =∂F

∂rij(D2v1, v1)(v1)xixj

, fi =∂c

∂xiv1, e = 〈Dcw1, Dv1〉 −

∂F

∂z(D2v1, v1),

δ = cw2(x− v2Dv2, t) − cw1(x− v2Dv2, t),

127


A(R, p, z, x, t) = F (D2v1 +R, v1 + z) − F (D2v1, v1)

− ∂F

∂z(D2v1, v1)z −

∂F

∂rij(D2v1, v1)rij

+ cw1(x− (v1 + z)(Dv1 + p), t) − cw1(x− v1Dv1, t)

+ 〈Dcw1(x− v1Dv1, t), Dv1〉z +∂cw1(x− v1Dv1, t)

∂xiv1 pi

and

a(p, x, t) =

−1

2

(〈2p, (Dv1(x, t) −Dg(x))〉 + |p|2

)on g = δ0

1

2

(〈2p, (Dv1(x, t) −Dg(x))〉 + |p|2

)on g = −δ0,

where we have used the fact that Dg is a unit normal to ∂V . Using the samearguments as those of Evans and Spruck [39, Lemma 5.3] (i.e. a Taylor expansion)and the fact that ‖v‖

C2+α,2+α2 (Q)

≤ 2R0, we get that

‖A‖Cα, α

2 (Q), ‖a‖

C1+α,1+α2 (Σ)

≤ C0R0‖v‖C2+α,2+α

2 (Q), (7.8)

where C0 does not depend on t0, R0. Using [39, Lemma 2.2], we then deduce that

‖v‖C2+α,2+α

2 (Q)≤ C1

(‖δ‖

Cα, α2 (Q)

+ ‖A‖Cα, α

2 (Q)+ ‖a‖

C1+α, 1+α2 (Σ)

),

where C1 does not depend on t0 and R0. This, together with (7.8) implies that

‖v‖C2+α,2+α

2 (Q)≤ 2C1‖δ‖Cα, α

2 (Q)(7.9)

as soon as R0 ≤ (4C0C1)−1. From now, let us fix on such an R0.

We now use the following lemma, the proof of which is postponed:

Lemma 7.6 (Estimate on the velocities).With the previous notation, there exists C independent of t0 such that if we set

w = w2 − w1, then we have, for t0 small enough,

‖δ‖W 1,1;∞(Q) ≤ C‖w‖W 1,1;∞(Q).

This implies in particular, also using the Hölder regularity of w and the fact thatwt(·, 0) = 0 = Dw(·, 0), that

‖δ‖Cα, α

2 (Q)≤ ‖δ‖W 1,1;∞(Q) ≤ C‖w‖W 1,1;∞(Q) ≤ Ct

α20 ‖w‖C2+α,2+α

2 (Q).

Using (7.9), we deduce that for t0 small enough,

‖v2 − v1‖C2+α,2+α

2 (Q)≤ 1

2‖w2 − w1‖

C2+α,2+α2 (Q)

.

128


This implies that Φ is a contraction, whence, using the Banach fixed point theorem,we deduce that there exists a unique solution v of (7.7).

Using Theorem 7.4, we finally obtain that, possibly reducing t0,

t ∈ [0, t0] 7→ Er(t) = (E0 \ V ) ∪ x ∈ V, v(x, t) ≥ 0

defines a smooth evolution with C3+α boundary starting from E0 with normal ve-locity

Vx,t = Hx,t + c0(·, t) ⋆ 1Er(t)(x) + c1(x, t).

This concludes the proof of Theorem 1.6.

We end with the proof of the last lemma:

Proof of Lemma 7.6. We begin by estimating the derivative of δ in time. Writingout the expression of ∂δ

∂t, we see that thanks to the fact that

‖v2‖C2+α, 2+α

2 (Q)≤ R0 + ‖g‖C2+α(V ),

the only difficult term to treat is ∂∂t

(cw2 − cw1). However we have, using Hadamard’sformula:

∂(cw2 − cw1)

∂t(x, t) =

∫

V

(c0)t(x− y, t)(1w2(·,t)≥0 − 1w1(·,t)≥0)(y) dy

+

∫

w2(·,t)=0(w2)t(y, t)c0(x− y, t)dHN−1(y)

−∫

w1(·,t)=0(w1)t(y, t)c0(x− y, t)dHN−1(y).

First, using Lemma 7.5, we have that∣∣∣∣∫

V

(c0)t(x− y, t)(1w2(·,t)≥0 − 1w1(·,t)≥0)(y)dy

∣∣∣∣ ≤ C‖c0‖W 1,∞([0,T ];L∞(RN ))‖w‖L∞(Q).

(7.10)For the other terms, we write

∫

w2(·,t)=0(w2)t(y, t)c0(x− y, t) dHN−1(y)

−∫

w1(·,t)=0(w1)t(y, t)c0(x− y, t) dHN−1(y) = I1 + I2,

where

I1 =

∫

w2(·,t)=0(w2)t(y, t)c0(x− y, t) dHN−1(y)

−∫

w2(·,t)=0(w1)t(y, t)c0(x− y, t) dHN−1(y)

129


and

I2 =

∫

w2(·,t)=0(w1)t(y, t)c0(x− y, t) dHN−1(y)

−∫

w1(·,t)=0(w1)t(y, t)c0(x− y, t) dHN−1(y).

We remark that|I1| ≤ C‖c0‖L∞(Q)‖wt‖L∞(Q), (7.11)

where the constant C is a bound on the perimeter of u(·, t) = 0, uniform for u ∈ Eand t ∈ [0, t0].

To estimate the term I2, we use a local parametrization. We choose local coor-dinates and r small enough such that if Br = BN−1

r (0), then

∂g

∂xN≥ 3

4in Br × [−r, r].

Now, recalling that

wi(·, 0) = g and ‖wi‖C2+α,2+α

2 (Q)≤ R0 + ‖g‖C2+α(V ),

we get that for t0 small enough (depending only on R0 and g),

∂wi

∂xN≥ 1

2in Br × [−r, r]. (7.12)

We fix t ≤ t0 and assume that wi(·, t) = 0∩(Br× [−r, r]) = (x′, fi(x′)), x′ ∈ Br.

Using a partition of unity, we will then recover the complete estimate. We defineε(x′) = f2(x

′) − f1(x′). For t0 small enough (depending only on R0 and g), we can

assume that|ε(x′)| ≤ 1

2(R0 + ‖g‖C2+α(V )). (7.13)

We then have

|I2| ≤C

∫

y′∈Br

∣∣∣√

1 + |Df1|2 c0(x′ − y′, xN − f1(y′), t)

−√

1 + |Df1 +Dε|2 c0(x′ − y′, xN − f1(y′) − ε(y′), t)

∣∣∣ dy′

≤C‖ε‖W 1,∞(Br),

where we have used the fact that c0 ∈ L∞([0, T ];W 1,∞(RN)) and where the constantC depends only on R0, g and c0.

Our goal now is just to estimate ‖ε‖W 1,∞(Br) with respect to ‖w‖L∞([0,t0],W 1,∞(V )).For simplicity of notation, we forget the dependence in time of w, w1 and w2. Werecall that

w1(x′, f1(x

′)) = 0 =w2(x′, f1(x

′) + ε(x′)) (7.14)

=w1(x′, f1(x

′) + ε(x′)) + w(x′, f1(x′) + ε(x′)).

130


Using a Taylor expansion, we get that

w1(x′, f1(x

′) + ε(x′)) =w1(x′, f(x′)) +

∂w1

∂xN(x′, f(x′)) ε(x′) + o(ε), (7.15)

where

‖o(ε)‖L∞ ≤ 1

2

∣∣∣∣∂2w1

∂x2N

∣∣∣∣ ‖ε‖2L∞ ≤ 1

4‖ε‖L∞,

thanks to (7.13) and the fact that

∣∣∣∣∂2w1

∂x2N

∣∣∣∣ ≤ R0 + ‖g‖C2+α(V ). We then deduce from

(7.14), (7.15) and (7.12) that

‖ε‖L∞ ≤ 4‖w‖L∞(Q). (7.16)

Differentiating (7.14) with respect to xi and using a Taylor expansion, we get asabove

‖εxi‖L∞ ≤ C

‖w‖L∞([0,t0],W 1,∞(V ))

| ∂w2

∂xN| ≤ 2C‖w‖L∞([0,t0],W 1,∞(V )).

Combining the last inequality with (7.16), we have

|I2| ≤ C‖w‖L∞([0,t0],W 1,∞(V )). (7.17)

Using (7.10), (7.11) and (7.17), we finally obtain∥∥∥∥∂δ

∂t

∥∥∥∥L∞(Q)

≤ C‖w‖W 1,1;∞(Q).

The estimates on ‖δ‖L∞(Q) and ‖Dδ‖L∞(Q) are easier (they use the regularity of c0),so we skip their proofs. This ends the proof of the lemma.

Acknowledgements. The authors would like to thank P. Cardaliaguet and R.Monneau for fruitful discussions during the preparation of this work, as well as Y.Giga for sending them some references. N. Forcadel was supported by the contractJC 1025 called “ACI jeunes chercheuses et jeunes chercheurs” of the French Ministryof Research (2003-2005). This work was partially supported by the ANR (AgenceNationale de la Recherche) through MICA project (ANR-06-BLAN-0082).

131

Chapitre 4

Uniqueness results for nonlocal

Hamilton-Jacobi equations

Ce chapitre est issu de l’article [18]. Il s’agit d’un travail en collaboration avec G.Barles, P. Cardaliaguet et O. Ley.

On s’intéresse dans ce chapitre à des équations eikonales non-locales et non-monotonesdécrivant l’évolution d’interfaces: plus précisément, on considère les deux équationsmodèle de la dynamique des dislocations et du systè me de FitzHugh-Nagumo (re-spectivement (3.2) et (3.7) dans l’introduction générale). Pour ce type d’équations,en général, seule l’existence de solutions faibles est connue; elle fait l’objet duchapitre 2. Nous proposons ici une nouvelle approche pour l’unicité de la solu-tion dans le cas de vitesses positives, c’est-à-dire où le front ne fait que grandir.Cette approche simplifie et étend des résultats existants pour la dynamique des dis-locations. Il s’agit en revanche du premier résultat d’unicité pour le système deFitzHugh-Nagumo. Les ingrédients principaux employés sont de nouvelles estima-tions de périmè tre pour le front, basées sur une propriété de cône intérieur uniformepour ces fronts.

Chapitre 4. Uniqueness results for nonlocal HJ equations

Abstract

We are interested in nonlocal eikonal equations describing the evolution

of interfaces moving with a nonlocal, non-monotone velocity. For these

equations, only the existence of global-in-time weak solutions is available

in some particular cases. In this chapter, we propose a new approach

for proving uniqueness of the solution when the front is expanding. This

approach simplifies and extends existing results for dislocation dynam-

ics. It also provides the first uniqueness result for a FitzHugh-Nagumo

system. The key ingredients are some new perimeter estimates for the

evolving fronts as well as some uniform interior cone property for these

fronts.

Key words and phrases: Front propagations, dislocation dy-namics, FitzHugh-Nagumo system, level-set approach, nonlocalHamilton-Jacobi equations, viscosity solutions, eikonal equation,L1−dependence in time, geometrical properties.

1 Introduction

In this chapter, we are interested in uniqueness results for different types of prob-lems which can be written as nonlocal Hamilton-Jacobi equations of the followingform:

ut = c[1u≥0](x, t)|Du| in RN × (0, T ),

u(x, 0) = u0(x) in RN ,(1.1)

where T > 0, the solution u is a real-valued function, ut and Du stand respectivelyfor its time derivative and space gradient and 1A is the indicator function of a setA. Finally u0 is a bounded, Lipschitz continuous function.

For any indicator function χ or more generally for any χ ∈ L∞(RN × [0, T ])with 0 ≤ χ ≤ 1 a.e., the function c[χ] depends on χ in a nonlocal way and, inthe main examples we have in mind, it is obtained from χ through a convolutiontype procedure (either only in space or in space and time). In particular, in ourframework, despite the fact that χ has no regularity either in x or in t, c[χ] willalways be Lipschitz continuous in x; on the contrary we impose no regularity withrespect to t. More precisely we always assume in what follows that, for any χ, thevelocity c = c[χ] satisfies

(H1) For all x ∈ RN , t 7→ c(x, t) is measurable and there exist C, c, c > 0 such that,for all x, y ∈ RN and t ∈ [0, T ],

|c(x, t) − c(y, t)| ≤ C|x− y|,0 < c ≤ c(x, t) ≤ c.

(1.2)

We will come back to this assumption later on.To give a first flavor of our main uniqueness results, we can point out the follow-

ing key fact: Equation (1.1) can be seen as the level-set equation associated to themotion of the front Γ(t) := x; u(x, t) = 0 with the nonlocal velocity c[1u(·,t)≥0].However, in the non-standard examples we consider, it is not only a nonlocal but

134

1. Introduction

also non-monotone geometrical equation; by non-monotone we mean that the inclu-sion principle, which plays a central role in the level-set approach, does not holdand, therefore, the uniqueness of solutions cannot be proved by standard viscositysolutions methods.

In fact, the few uniqueness results which exist in the literature (see below) relyon L1 type estimates on the solution; this is natural since one has to connect thecontinuous function u and the indicator function 1u≥0. The main estimates concernmeasures of sets of the type x; a ≤ u(x, t) ≤ b for a, b close to 0. Whether or notthe aforementioned estimate has to be uniform in time, or of integral type, stronglydepends on the properties of the convolution kernel. In order to emphasize this fact,we are going to concentrate on two model cases: the first one is a dislocation typeequation (see Section 3) in which the kernel belongs to L∞, while the second oneis related to the FitzHugh-Nagumo system arising in neural wave propagation or inchemical kinetics, in which the kernel is essentially the kernel of the Heat Equation(see Section 4). In that case, it is not in L∞. The fact that the convolution kernelis, or is not, bounded is indeed the key difference here.

Before going further, let us give some references: for the first model case (disloca-tion type equations, see (3.1) below), we refer the reader to Barles, Cardaliaguet, Leyand Monneau [16] where general results are provided for these equations. We pointout – and we will come back to this fact later – that uniqueness in the non-monotonecase was first obtained by Alvarez, Cardaliaguet and Monneau [2] and then by Barlesand Ley [20] using different arguments; we also refer to Rodney, Le Bouar and Finel[66] for the physical background on these equations. The FitzHugh-Nagumo systemhas been investigated in particular by Giga, Goto and Ishii [48], and by Soravia andSouganidis [72]. They provided a notion of weak solution for this system (see (4.1)below) and proved existence of such weak solutions. Soravia and Souganidis [72]also studied the connections with the phase field model (a reaction-diffusion systemwhich leads to such a front propagation model). However the uniqueness questionhad been left open.

Let us return to the key steps to prove uniqueness for (1.1). A major issue is theproperties of the solutions of eikonal equations of the form

ut = c(x, t)|Du| in RN × (0, T ), (1.3)

where c is a continuous function, satisfying suitable assumptions. Of course, suchpartial differential equations appear naturally when considering 1u≥0 as an a priorigiven function in (1.1). We recall that (1.3) is related via the level-set approach tothe motion of fronts with a (x, t)-dependent normal velocity c(x, t), and to deal withcompact fronts and simplify matter, we assume that the initial datum satisfies thefollowing conditions: the subset u0 > 0 is non-empty and there exists R0 > 0 suchthat

u0 = −1 in RN \ B(0, R0). (1.4)

This implies, in particular, that the initial front Γ0 = u0 = 0 is a non-emptycompact subset of B(0, R0).

135


Assumption (H1) ensures existence and uniqueness of a solution to (1.3), butwe also assume that the function c = c[χ] is everywhere positive, together with thefollowing regularity assumption on Γ0:

(H2) There exists η0 > 0 such that

−|u0(x)| − |Du0(x)| + η0 ≤ 0 in RN in the viscosity sense.

The above assumptions imply that the set u = 0 has zero Lebesgue measure (cf.Ley [53]) which is an important property for our arguments. Indeed [16] providesthe counter-example of a nonlocal equation (even in a (quasi) monotone case) wherethe fattening phenomenon leads to a non-uniqueness feature. In addition to thisnon-fattening property, a key consequence of (H1)-(H2) is a lower bound on thegradient Du on the set x; |u(x, t)| ≤ η for some small enough η > 0 (cf. [53]).

We now concentrate on the estimates of the measures of level-sets of the forma ≤ u(·, t) ≤ b where −η ≤ a < b ≤ η. We first note that such estimates arerelated with perimeter estimates of the α level-sets of u for α close to 0 (typically|α| < η): indeed, combining the co-area formula with the lower bound on thegradient of the solution, we obtain that

∫

RN

1a≤u(·,t)≤b(x) dx =

∫ b

a

∫

u(·,t)=s|Du|−1 dHn−1ds

≤ b− a

ηsup

a≤s≤bP (u(·, t) = s),

(1.5)

where η is the lower bound for |Du| on the set x; |u(x, t)| ≤ η.In [2] and [20], perimeter estimates for the α level-sets of u were obtained by using

bounds on the curvature of these sets. Although this approach was powerful, itsdrawback is to require strong assumptions on the dependence of c[χ] on x (typicallya C1,1 regularity). Unfortunately such strong regularity does not always hold: forinstance it is not the case for the FitzHugh-Nagumo system.

The key contribution of this work is to provide L1([0, T ]) or L∞([0, T ]) estimatesof the volume of the set a ≤ u(·, t) ≤ b (or, almost equivalently, of the perimeterof the α level-sets of u) in situations where the velocity c[χ] is less regular in x. Asa consequence we are able to prove new uniqueness results.

For the dislocation dynamics model, our approach allows to relax the assump-tions of [2] and [20] on the data. The proofs are also simpler, requiring only L1([0, T ])estimates and a mild regularity (c[χ] is merely measurable in time and Lipschitz con-tinuous in space). So the main conclusion here is that “soft” estimates are sufficientprovided the convolution kernel is in L∞.

On the contrary, for the FitzHugh-Nagumo system, where the convolution ker-nel is unbounded, these L1-estimates are not sufficient anymore and the proof ofuniqueness requires heavy L∞-estimates on the perimeter instead. These estimatesare obtained by establishing, through optimal control type arguments, that the setx; u(x, t) ≥ 0 satisfies a uniform “interior cone property”, from which we deduce(explicit) estimates on the perimeter.

136

2. Definition of weak solutions to (1.1)

This chapter is organized as follows: in Section 2, we recall the notion of weaksolution for (1.1) introduced in [16] and already used in Chapter 2. In Section 3we prove uniqueness of the solution for the dislocation type equation, while we dealwith the FitzHugh-Nagumo case in Section 4. The main technical results of thischapter are gathered in Section 5: we recall here some useful results for the eikonalequation (1.3), we show the interior cone property and deduce the uniform perimeterestimates.

Notation. In what follows, |·| denotes the standard Euclidean norm on RN , B(x,R)(resp. B(x,R)) is the open (resp. closed) ball of radius R centered at x ∈ RN . Wedenote the essential supremum of f ∈ L∞(RN) or f ∈ L∞(RN × [0, T ]) by ‖f‖∞.Finally, Ln and Hn denote, respectively, the n-dimensional Lebesgue and Hausdorffmeasures.

2 Definition of weak solutions to (1.1)

We will use the following definition of weak solutions introduced in [16] andstudied in Chapter 2:

Definition 2.1. Let u : RN × [0, T ] → R be a continuous function. We say that uis a weak solution of (1.1) if there exists χ ∈ L∞(RN × [0, T ]; [0, 1]) such that

1. u is a L1-viscosity solution ofut(x, t) = c[χ](x, t)|Du(x, t)| in RN × (0, T ),u(·, 0) = u0 in RN .

(2.1)


1u(·,t)>0 ≤ χ(·, t) ≤ 1u(·,t)≥0 a.e. in RN .


1u(·,t)>0 = 1u(·,t)≥0 a.e. in RN .

We refer to the general introduction or [16, Appendix] for basic definition andproperties of L1-viscosity solutions and to [50, 61, 62, 27, 26] for a complete presen-tation of the theory.

3 Model problem 1: dislocation type equations

In this section, we consider equations arising in dislocations theory (see thegeneral introduction and Chapter 2 or [17]) where, for all χ ∈ L∞(RN × [0, T ]), c[χ]is defined by

c[χ](x, t) = c0(·, t) ⋆ χ(·, t)(x, t) + c1(x, t) in RN × (0, T ), (3.1)

137


where c0, c1 are given functions, satisfying suitable assumptions which are describedlater on, and ⋆ stands for the usual convolution in RN with respect to the spacevariable x. Our main result below applies to slightly more general cases but themain interesting points appear for this model case.

We refer to [16] for a complete description of the characteristics and difficultiesconnected to (1.1) in this case; as recalled in the introduction, this equation isnot only nonlocal but it is also, in general, non-monotone, which means that themaximum principle (or here, inclusion principle) does not hold and one cannot applydirectly viscosity solutions’ theory. Roughly speaking, a (more or less) direct use ofviscosity solutions’ theory requires that c0 ≥ 0 in RN × (0, T ), an assumption whichis not natural in the dislocations’ framework.

We use the following assumptions on c0 and c1:(D) (i) c0 ∈ C0([0, T ];L1

(RN)), c1 ∈ C0(RN × [0, T ]; R).

(ii) For any t ∈ [0, T ], c0(·, t) is locally Lipschitz continuous, and there exists aconstant C > 0 such that ‖Dc0‖L∞([0,T ];L1(RN )) ≤ C.

(iii) There exists a constant C > 0 such that, for any x, y ∈ RN and t ∈ [0, T ],

|c1(x, t) − c1(y, t)| ≤ C|x− y|.

(iv) There exist c, c > 0 such that, for any x ∈ RN and t ∈ [0, T ],

|c0(x, t)| ≤ c ,

0 < c ≤ −‖c0(·, t)‖L1(RN ) + c1(x, t) ≤ ‖c0(·, t)‖L1(RN ) + c1(x, t) ≤ c .

This assumption ensures that the velocity c[χ] in (3.1) satisfies (H1) with constantsindependent of 0 ≤ χ ≤ 1 with compact support in some fixed ball (see Step 1 in theproof of Theorem 3.1). Assumption (D) can be slightly relaxed (and in particularlocalized) using that the front remains in a bounded region of RN . Note that, incontrast to [16], we do not assume that c0, c1 are C1,1 (or semiconvex).

We provide a direct proof of uniqueness for the solution of the dislocation equa-tion (1.1); we recall that the existence of weak solutions is obtained in [16], and inChapter 2 or [17], and that, in our case, thanks to (H2) and the fact that c[χ] ≥ 0for all 0 ≤ χ ≤ 1, the weak solutions are classical solutions since the set u = 0has zero Lebesgue measure by the result of [53].

Theorem 3.1. Suppose that c0, c1 satisfy (D) and that u0 is a Lipschitz continu-ous function satisfying (H2) and such that (1.4) holds. Then (1.1) has a unique(Lipschitz) continuous viscosity solution in RN × [0, T ].

Proof of Theorem 3.1.Step 1. Uniform bounds for the velocity. By (D) and Lemma 5.3, we know that theset u(·, t) ≥ 0 remains in a fixed ball B(0, R0 + cT ) of RN . Then, for any subsetA of B(0, R0 + cT ), c[1A] satisfies (H1) with constants which are uniform in A.

Step 2. L∞-estimate. If u1, u2 are two solutions of (1.1), for 0 < τ ≤ T, we set


|u1(x, t) − u2(x, t)|.

138

3. Model problem 1: dislocation type equations

Since u0 is Lipschitz continuous and 0 ≤ c[1ui≥0] ≤ c (i = 1, 2), for τ smallenough, we have δτ < η/2 where η is obtained by applying Theorem 5.1 to the ui’s.By Lemma 5.2, we have


∫ τ

0

‖(c[1u1≥0] − c[1u2≥0])(·, t)‖∞ dt

≤ ‖Du0‖∞eCτ

∫ τ

0

‖c0(·, t) ⋆ (1u1(·,t)≥0 − 1u2(·,t)≥0)‖∞ dt

≤ c ‖Du0‖∞eCT

∫ τ

0

∫

RN

|1u1≥0 − 1u2≥0|(x, t) dxdt, (3.2)

using the L∞-bound ‖c0‖∞ ≤ c.

Step 3. L1-estimate. By definition of δτ , we have

|1u1≥0 − 1u2≥0| ≤ 1−δτ≤u1≤0 + 1−δτ≤u2≤0 in RN × [0, τ ].

Using the key L1-estimate Proposition 5.5, we get∫ τ

0

∫

RN

|1u1≥0 − 1u2≥0|(x, t) dxdt ≤2δτηcψτ ,

where we have set

ψτ = LN(u0 ≥ −δτ − c‖Du0‖∞τ) −LN(u0 ≥ 0) .

Step 4. Uniqueness on [0, τ ] for small τ. Using this information in (3.2) yields

δτ ≤ 2c

ηc‖Du0‖∞eCTψτδτ ,

that is,

δτ ≤ Lψτ δτ ,

where L = L(T, c, c, C, η, ‖Du0‖∞) is a constant. Since the 0 level-set of u0 has zeroLebesgue measure from assumption (H2), we have ψτ → 0 as τ → 0. Therefore,for τ small enough, Lψτ < 1 and necessarily δτ = 0. It follows that u1 = u2 onRN × [0, τ ].

Step 5. Uniqueness on the whole time interval. Step 4 gives the uniqueness for smalltimes, but then we can consider

τ = supτ ∈ [0, T ]; u1 = u2 on RN × [0, τ ].

In fact, by continuity of u1 and u2, τ is a maximum. If τ < T , then we can repeatthe above proof from time τ instead of time 0 and obtain a contradiction. This is, infact, rather straightforward since u(·, τ) satisfies the same properties as u0. Finally,we get that τ = T and the proof is complete.

139


4 Model problem 2: a FitzHugh-Nagumo type sys-

tem

We are now interested in the following system presented in the general introduc-tion, and studied in Chapter 2, or [17]:

ut = α(v)|Du| in RN × (0, T ),

vt − ∆v = g+(v)1u≥0 + g−(v)(1 − 1u≥0) in RN × (0, T ),

u(·, 0) = u0, v(·, 0) = v0 in RN ,

(4.1)

which is obtained as the asymptotics as ε → 0 of the following FitzHugh-Nagumosystem arising in neural wave propagation or chemical kinetics (cf. [72]):

uε



in RN × (0, T ), where, for (u, v) ∈ R

2,f(u, v) = u(1 − u)(u− a) − v (0 < a < 1),

g(u, v) = u− γv (γ > 0).

The functions α, g+ and g− appearing in (4.1) are Lipschitz continuous functionson R associated with f and g.

System (4.1) corresponds to a front Γ(t) = u(·, t) = 0 moving with normalvelocity α(v), the function v being itself the solution of an interface reaction-diffusionequation depending on the regions separated by Γ(t). The u-equation in (4.1) canbe written as (1.1) although the dependence of c in 1u≥0 is less explicit than in thefirst model case. More precisely, for χ ∈ L∞(RN × [0, T ]; [0, 1]), let v be the solutionof

vt − ∆v = g+(v)χ+ g−(v)(1 − χ) in RN × (0, T ),v(·, 0) = v0 in RN .

(4.2)

Then Problem (4.1) reduces to (1.1) with c[χ](x, t) = α(v(x, t)).Under the additional assumption that α > 0 in R, we prove uniqueness of so-

lutions to (4.1), or equivalently (1.1). We make the following assumptions on thedata:

(F) (i) α is Lipschitz continuous on R and there exist c, c, C > 0 such that, forall r, r′ ∈ R,

c ≤ α(r) ≤ c

|α(r)−α(r′)| ≤ C|r − r′|.(ii) g+ and g− are Lipschitz continuous on RN , and there exist g and g in R

such thatg ≤ g−(r) ≤ g+(r) ≤ g for all r in R.

(iii) v0 is bounded and of class C1 with ‖Dv0‖∞ < +∞.

140

4. Model problem 2: a FitzHugh-Nagumo type system

We also make the following additional assumption on the initial datum u0:

(H3) u0 is Lipschitz continuous and satisfies (1.4) with K0 := u0 ≥ 0 which isthe closure of a non empty bounded open subset of R

N with C2 boundary.

Our result is the following:

Theorem 4.1. Under assumptions (F), (H2) and (H3), the system (4.1) has aunique solution.

We recall that the existence of weak solutions is obtained in [48, 72], and inChapter 2 or [17]. Moreover, since α > 0 in R, weak solutions are classical thanksto the results of [53]. Before giving the uniqueness proof, we start by a preliminaryresult on the inhomogeneous heat equation.

4.1 Classical estimates for the inhomogeneous heat equation.

We first gather some regularity results for the solutions of the heat equation (4.2).The explicit resolution of this equation shows that for any (x, t) ∈ RN × (0, T ),

v(x, t) =

∫

RN


+

∫ t

0

∫

RN



G(y, s) =1

(4πs)N/2e−

|y|2

4s . (4.3)

We recall the result of Chapter 2, Lemma 4.5, that we easily deduce from thisrepresentation formula.

Lemma 4.2. Assume that (F) holds. For χ ∈ L∞(RN × [0, T ]; [0, 1]), let v be theunique solution of (4.2). Set γ = max|g|, |g|. Then there exists a constant kN

depending only on N such that(i) v is uniformly bounded: for all (x, t) ∈ RN × [0, T ],

|v(x, t)| ≤ ‖v0‖∞ + γt.

(ii) v is continuous on RN × [0, T ].

(iii) For any t ∈ [0, T ], v(·, t) is of class C1 in RN .(iv) For all t ∈ [0, T ], x, y ∈ RN ,

|v(x, t) − v(y, t)| ≤ ( ‖Dv0‖∞ + γkN

√t) |x− y|.

(v) For all 0 ≤ s ≤ t ≤ T, x ∈ RN ,

|v(x, t) − v(x, s)| ≤ kN(‖Dv0‖∞ + γkN

√s)

√t− s+ γ(t− s).

In particular the velocity c[χ] (given here by α(v)) is bounded, continuous onRN × [0, T ] and Lipschitz continuous in space, uniformly with respect to χ.

141


4.2 Proof of Theorem 4.1

Step 1. Uniform bounds for the velocity. As explained above, for any measurablesubset A of RN , the velocity c[1A] in (1.1) satisfies (H1) with constants which areuniform in A: for all x, x′ ∈ RN , t, t′ ∈ [0, T ],

c ≤ c[1A] ≤ c

|c[1A](x, t) − c[1A](x′, t)| ≤ C|x− x′|,|c[1A](x, t) − c[1A](x, t′)| ≤ kN C|t− t′|1/2 + γ |t− t′|,

with C := C(‖Dv0‖∞+γkN

√T ). By Lemma 5.3, it follows that the set u(·, t) ≥ 0

remains in a fixed ball B(0, R0 + cT ) of RN .

Step 2. First estimate (eikonal equation). We start as in the proof of Theorem 3.1.Let u1, u2 be two solutions of (1.1) and v1, v2 be the solutions of (4.2) associatedwith 1u1≥0, 1u2≥0 respectively. For 0 < τ ≤ T, we set


|u1(x, t) − u2(x, t)|

and we choose τ small enough in order that δτ < η/2 where η is given by applyingTheorem 5.1 to the ui’s. By Lemma 5.2, we have


∫ τ

0

‖(c[1u1≥0] − c[1u2≥0])(·, t)‖∞ dt

≤ ‖Du0‖∞eCτ

∫ τ

0

‖(α(v1) − α(v2))(·, t)‖∞ dt

≤ C‖Du0‖∞eCT

∫ τ

0

‖(v1 − v2)(·, t)‖∞ dt. (4.4)

It remains to estimate ‖(v1 − v2)(·, t)‖∞.Step 3. Second Estimate (heat equation). The function v = v1 − v2 solves

vt − ∆v =(1u1≥0 − 1u2≥0)(g+(v1) − g−(v1))

+ 1u2≥0(g+(v1) − g+(v2)) + (1 − 1u2≥0)(g

−(v1) − g−(v2))

in RN ×(0, T ). Since g+ and g− are Lipschitz continuous, say with Lipschitz constant

D, we have

|1u2≥0(g+(v1) − g+(v2)) + (1 − 1u2≥0)(g

−(v1) − g−(v2))| ≤ D|v|.

Moreover, using (F), we have

|1u1≥0 − 1u2≥0| |g+(v1) − g−(v1)| ≤ (g − g)|1u1≥0 − 1u2≥0|.

This implies that both v and −v are viscosity subsolutions of

wt − ∆w −D|w| = (g − g)|1u1≥0 − 1u2≥0| in RN × (0, T ),

142


whence |v| = maxv,−v is also a subsolution as the maximum of two subsolutions.Therefore we have

|v|t − ∆|v| −D|v| ≤ (g − g)|1u1≥0 − 1u2≥0| in RN × (0, T ).

In particular the function w : (x, t) 7→ e−Dt |v(x, t)| satisfies

wt − ∆w ≤ (g − g) e−Dt |1u1≥0 − 1u2≥0| in RN × (0, T ).

By the comparison principle, since w(·, 0) = 0, we have for any (x, t) ∈ RN × [0, τ ],

w(x, t) ≤∫ t

0

∫

RN

G(x− y, t− s) (g − g) e−Ds |1u1≥0 − 1u2≥0|(y, s) dyds.

Using the definition of δτ , we have

|1u1≥0 − 1u2≥0|(y, s) ≤ 1−δτ≤u1<0 + 1−δτ≤u2<0.

This implies that for any (x, t) ∈ RN × [0, τ ],

|v1(x, t) − v2(x, t)| (4.5)

≤ (g − g) eDT

∫ t

0

∫

RN

G(x− y, t− s)(1−δτ≤u1<0 + 1−δτ≤u2<0

)(y, s) dyds.

For simplicity, we set B = B(0, 1) and for i = 1, 2,

Ki(t) = ui(·, t) ≥ 0.

Step 4. We claim that −δτ ≤ ui(·, t) < 0 ⊂ (Ki(t) + δτB/η) \ Ki(t) where η isgiven by (5.2). Indeed let x ∈ RN be such that −δτ ≤ ui(x, t) < 0. Using (5.2),and since we chose δτ small enough in Step 2, Lemma 5.4 implies that there existsy ∈ B(x, δτ/η) such that ui(y, t) ≥ ui(x, t) + δτ ≥ 0. This proves the claim.

Step 5. Use of an interior cone property for the Ki(t)’s. From (4.5) and Step 4, weobtain the estimation

|v1(x, t) − v2(x, t)| (4.6)

≤ (g − g) eDT

∫ t

0

∫

RN

G(x− y, t− s) (1E1(t)(y) + 1E2(t)(y)) dyds,

where Ei(t) = (Ki(t) + δτB/η) \Ki(t) for i = 1, 2.We are now going to use the fact that the sets K1(t) = u1(·, t) ≥ 0 and

K2(t) = u2(·, t) ≥ 0 have the interior cone property (see Definition 5.7) for allt ∈ [0, T ], for some parameters ρ and θ independent of t:

Lemma 4.3. There exist ρ and θ depending only on the data (N , T , u0, v0, α,g+ and g−) such that 0 < ρ < θ < 1 and Ki(t) has the interior cone property ofparameters ρ and θ for all t ∈ [0, T ].

143


This lemma is an application of Theorem 5.9 below (see Section 5.4), the assumptionsof which are satisfied for u1 and u2 because of Step 1. As a consequence, we can usethe following lemma which is proved in Section 4.3 below:

Lemma 4.4. Let K(t)t∈[0,T ] ⊂ B(0, R) × [0, T ] (R > 0) be a bounded family ofcompact subsets of R

N having the interior cone property of parameters ρ and θ with0 < ρ < θ < 1, and let us set, for any x ∈ RN , t ∈ [0, T ] and r ≥ 0,

φ(x, t, r) =

∫ t

0

∫

RN

G(x− y, t− s)1K(s)+rB(y) dyds.

Then for any r0 > 0, there exists a constant Λ0 = Λ0(N,R, r0, ρ, θ/ρ) such that forany x ∈ RN , t ∈ [0, 1] and r ∈ [0, r0],

|φ(x, t, r) − φ(x, t, 0)| ≤ Λ0 r.

We apply this lemma to the Ki(t)’s which satisfy the assumptions with R = R0 +cTby Step 1 and since we can assume that τ < 1. From (4.4) and (4.6), we finallyobtain that

δτ ≤ L τ δτ

where L = L(N, T, u0, v0, α, g, g). Choosing τ such that Lτ < 1, we obtain δτ = 0.We conclude as in the proof of Theorem 3.1.

4.3 Proof of Lemma 4.4

For any x ∈ RN , t ∈ [0, 1] and r ∈ [0, r0],

φ(x, t, r) − φ(x, t, 0) =

∫ t

0

∫

RN

G(x− y, t− s)(1K(s)+rB − 1K(s)

)(y) dyds.

Let dK(s) denote the signed distance function to K(s), namelydK(s)(x) = dK(s)(x) if x /∈ K(s),

dK(s)(x) = −d∂K(s)(x) if x ∈ K(s),

where, for any closed non-empty set A ⊂ RN , dA is the usual distance to A. Then1K(s)+rB − 1K(s) = 10<dK(s)≤r, so that

φ(x, t, r) − φ(x, t, 0) =

∫ t

0

∫

0<dK(s)≤rG(x− y, t− s) dyds.

Since dK(s) is Lipschitz continuous with |DdK(s)| = 1 almost everywhere in the set0 < dK(s) ≤ r, the coarea formula (see [38]) shows that

φ(x, t, r) − φ(x, t, 0) =

∫ t

0

∫ r

0

∫

dK(s)=σG(x− y, t− s) dHN−1(y)dσds

=

∫ t

0

∫ r

0

dσ

∫

dK(s)=σ

1

(4π(t− s))N/2e−

|x−y|2

4(t−s) dHN−1(y)ds.

144


The change of variable z = y−x√t−s

in this last integral yields

φ(x, t, r) − φ(x, t, 0) =1

(4π)N/2

∫ r

0

dσ

∫ t

0

1√t− s

∫

ζs,σ

e−|z|2

4 dHN−1(z)ds,

where we have set

ζs,σ =

y − x√t− s

; dK(s)(y) = σ

.

For some R(s) to be precised later, we split∫

ζs,σe−

|z|2

4 dHN−1(z) in two parts, one

in BR(s) = B(0, R(s)), and one in BcR(s). First, for any s ∈ [0, t) and σ > 0,

∫

ζs,σ∩BR(s)

e−|z|2

4 dHN−1(z)

≤ HN−1(ζs,σ ∩BR(s))

≤ Λ(N, ρ, θ/ρ)ωN (R(s) + ρ/4)N

≤ Λ(N, ρ, θ/ρ)ωN (R(s) + 1)N

where Λ(N, ρ, θ/ρ) is the constant given by Theorem 5.8, and ωN is the volume ofthe unit ball of RN . Indeed, for any s ∈ [0, t) and σ > 0,

ζs,σ = ∂

y − x√t− s

; dK(s)(y) < σ

,

and these sets inherit the interior cone property from K(s), with parameters greaterthan ρ and θ. Besides

∫

ζs,σ∩BcR(s)

e−|z|2

4 dHN−1(z)

≤ e−R(s)2

4 HN−1(ζs,σ)

≤ e−R(s)2

41

(t− s)N−1

2

HN−1(dK(s) = σ

)

≤ e−R(s)2

41

(t− s)N−1

2

Λ(N, ρ, θ/ρ)ωN (R + r0 + ρ/4)N

≤ e−R(s)2

41

(t− s)N−1

2

Λ(N, ρ, θ/ρ)ωN (R + r0 + 1)N ,

becausedK(s) ≤ σ

⊂ BR+r0 for any s ∈ [0, τ ] and r ∈ [0, r0]. This estimate also

comes from Theorem 5.8 for the same reason as above. Thus we have proved theexistence of a constant

Λ1 = Λ1(N,R, r0, ρ, θ/ρ) =1

(4π)N/2Λ(N, ρ, θ/ρ)ωN (R + r0 + 1)N

such that for any x ∈ RN , t ∈ [0, 1] and r ∈ [0, r0],

|φ(x, t, r) − φ(x, t, 0)| ≤ Λ1 r

∫ t

0

1√t− s

((R(s) + 1)N +

e−R(s)2

4

(t− s)N−1

2

)ds. (4.7)

145


Choosing R(s) =√

−2(N − 1)ln(t− s), so that e−R(s)2

4 = (t−s)N−12 , we can estimate

the right-hand side of (4.7) as follows:

∫ t

0

1√t− s

((R(s) + 1)N +

e−R(s)2

4

(t− s)N−1

2

)ds

≤∫ 1

0

(|2(N − 1)ln(u)|1/2 + 1)N + 1√u

du =: I(N) .

We deduce the existence of the constant

Λ0 = Λ0(N,R, r0, ρ, θ/ρ) = Λ1I(N)

such that for any x ∈ RN , t ∈ [0, 1] and r ∈ [0, r0],

|φ(x, t, r) − φ(x, t, 0)| ≤ Λ0 r.

5 Eikonal equation, interior cone property and perime-

ter estimates

5.1 Some results on the classical eikonal equation

In this section, we collect several properties of the eikonal equation (1.3). Wefirst recall the following result on existence and regularity:

Theorem 5.1 (see [53]).

(i) Under assumption (H1), equation (1.3) has a unique continuous viscosity so-lution u. If u0 is Lipschitz continuous, then u is Lipschitz continuous and, foralmost all x ∈ RN , t ∈ [0, T ],

|Du(x, t)| ≤ eCT‖Du0‖∞ , |ut(x, t)| ≤ c eCT‖Du0‖∞ .

(ii) Assume that u0 is Lipschitz continuous and that (H1) and (H2) hold. Thenthere exist γ = γ(C, c, η0) > 0, η = η(C, c, η0) > 0 such that the viscositysolution u of (1.3) satisfies in the viscosity sense

−|u(x, t)| − eγt

4|Du(x, t)|2 + η ≤ 0 in R

N × [0, T ]. (5.1)

We refer the reader to [53] for the proof of this result. Let us mention that(H1) implies that p ∈ RN 7→ c(x, t)|p| is convex for every (x, t) ∈ RN × [0, T ],which is a key assumption to prove (ii). We remark that, in (ii), u is Lipschitzcontinuous because the assumptions of (i) are satisfied. Therefore u is differentiablea.e. in RN × [0, T ] and (5.1) holds a.e. in RN × [0, T ]. Part (ii) gives a lower-bound

146

5. Eikonal equation, interior cone property and perimeter estimates

gradient estimate for u near the front (x, t) ∈ RN × [0, T ]; u(x, t) = 0. Indeed, if

|u(x, t)| ≤ η/2, then

−|Du(x, t)| ≤ −√

2η e−γT/2 =: −η < 0 (5.2)

in the viscosity sense and almost everywhere.We continue by giving an upper-bound for the difference of two solutions with

different velocities ci.

Lemma 5.2 (see [20]). For i = 1, 2, let ui ∈ C0(RN × [0, T ]) be a solution of

(ui)t = ci(x, t)|Dui| in RN × [0, T ],

ui(x, 0) = u0(x) in RN ,

where ci satisfies (H1) and u0 is Lipschitz continuous. Then, for any t ∈ [0, T ],

‖(u1 − u2)(·, t)‖∞ ≤ ‖Du0‖∞eCt

∫ t

0

‖(c1 − c2)(·, s)‖∞ ds.

Finite speed of propagation implies a uniform bound for compact fronts governedby eikonal equations:

Lemma 5.3 (see [20]). Suppose that (H1) holds and that u0 is Lispchitz continuousand satisfies (1.4). Let u be the viscosity solution of (1.3) with initial condition u0.Then, for all t ∈ [0, T ],

u(·, t) ≥ 0 ⊂ B(0, R0 + ct).

Lemma 5.4 (viscosity increase principle, see [20]). Let w ∈ C0(RN) be such thatthere exists two positive constants η and η such that on the set |w| ≤ η, we have

−|Dw(x)| ≤ −η

in the viscosity sense. If δ < η and x ∈ −δ ≤ w ≤ δ, then

supB(x,δ/η)

w ≥ w(x) + δ.

We refer the reader to [20] for the proofs of these results.

5.2 Estimates on the measure of level-sets for solutions of

(1.3).

Now we turn to the key estimates on the measure of small level-sets of thesolution of the eikonal equation (1.3). For every −η/2 ≤ a < b ≤ η/2, we considerthe function ϕ : R → R+, depending on a and b, such that ϕ = 0 on (−∞, a),ϕ′(t) = (b − a)−1 in (a, b) and ϕ = 1 on [b,+∞). In fact, ϕ is chosen in such a waythat (b− a)ϕ′ is the indicator function of [a, b]. We omit to write the dependence ofϕ with respect to a, b for the sake of simplicity of notation.

147


Proposition 5.5. Assume (H1), (H2), and suppose that u0 ≥ 0 is a compactsubset of RN . Let −η/2 ≤ a < b ≤ η/2 where η is defined in (5.1) and let u be theunique Lipschitz continuous viscosity solution of (1.3). Then, for any 0 < τ ≤ T ,

∫ τ

0

∫

RN

1a≤u≤b(x, t) dxdt ≤b− a

ηc

∫

RN

[ϕ(u(x, τ)) − ϕ(u(x, 0))] dx, (5.3)

where η is defined in (5.2). It follows∫ τ

0

∫

RN

1a≤u≤b(x, t) dxdt ≤b−aηc

[LN(u(·, τ) ≥ a) − LN(u(·, 0) ≥ b)

]dx, (5.4)

and ∫ τ

0

∫

RN

1a≤u≤b(x, t) dxdt (5.5)

≤ b− a

ηc

[LN (u(·, 0) ≥ a− c‖Du0‖∞τ) −LN (u(·, 0) ≥ b)

].

Remark 5.6. The above Proposition is related with results obtained in Chapter 1 or[58] for the eikonal equation with a velocity which changes sign.

Proof of Proposition 5.5. By definition of ϕ,∫ τ

0

∫

RN

1a≤u≤b(x, t) dxdt =

∫ τ

0

∫

RN

(b− a)ϕ′(u(x, t)) dxdt.

Using the fact that −η/2 ≤ a < b ≤ η/2 and the definition of η in (5.2), we canestimate the right-hand side by

∫ τ

0

∫

RN

(b− a)ϕ′(u(x, t))c(x, t)

c

|Du|η

dxdt,

since c ≤ c on RN × [0, T ] and |Du| ≥ η a.e. on the set |u| ≤ η/2, thanks to (5.2).Besides, using the equation, we have the following equality:

b− a

ηc

∫ τ

0

∫

RN

ϕ′(u(x, t))c(x, t)|Du| dxdt =(b− a)

ηc

∫ τ

0

∫

RN

(ϕ(u(x, t)))t dxdt,

and (5.3) follows by applying Fubini’s Theorem and integrating. Inequality (5.4)follows easily by taking into account the form of ϕ. To deduce (5.5), it is sufficientto note that, since u0+c‖Du0‖∞t is a supersolution of (1.3), we have, by comparison,u(x, t) ≤ u0(x) + c‖Du0‖∞t in RN × [0, T ].

5.3 Estimate of the perimeter of sets with the interior cone

property

Definition 5.7. Let K be a compact subset of RN . We say that K has the interior

cone property of parameters ρ and θ if 0 < ρ < θ and if, for any x ∈ ∂K, thereexists some ν ∈ SN−1 such that the set

Cρ,θν,x := x+ [0, θ]B(ν, ρ/θ)

= x+ λν + λρθξ; λ ∈ [0, θ], ξ ∈ B(0, 1)

is contained in K.

148


Theorem 5.8. Let K be a compact subset of RN having the interior cone property

of parameters ρ and θ. Then there exists a positive constant Λ = Λ(N, ρ, θ/ρ) suchthat for all R > 0,

HN−1(∂K ∩ B(0, R)) ≤ ΛLN(K ∩ B(0, R+ ρ/4)). (5.6)

z

θ

ρ

slope√

(θ/ρ)2 − 1

ρ

K

Cρ,θν,z

ν

Figure 4.1: Cρ,θν,z : interior cone at z of parameters ρ, θ and axis ν.

Proof.Step 1. Restriction to a finite number of axes for the interior cones. We first observethat if z ∈ ∂K and Cρ,θ

ν,z ⊂ K, then for all ν ′ ∈ SN−1 verifying |ν − ν ′| ≤ ρ/(2θ), we

have Cρ/2,θν′,z ⊂ K. By compactness of SN−1, we can cover SN−1 with the traces on

SN−1 of at most

p :=β(N)

(ρ/(2θ))N−1

balls of radius ρ/(2θ) centered at νi ∈ SN−1, for some positive constant β(N) and1 ≤ i ≤ p. Therefore, for any z ∈ ∂K, there exists 1 ≤ i ≤ p such that Cρ/2,θ

νi,z ⊂ K.

Step 2. Local study of points of the boundary with the same interior cone axis. Wefix 1 ≤ i ≤ p and set Ai = z ∈ ∂K; Cρ/2,θ

νi,z ⊂ K. Up to a rotation of K, we canassume that νi = (0, . . . , 0,−1) =: ν. Let us fix z ∈ Ai, that we write z = (x, y)with x ∈ RN−1 and y ∈ R. Let us set V = BN−1(x, ρ/4) × (y − θ/2, y + θ/2) and

Di = V ∩⋃

(x′,y′)∈Ai∩V

Cρ/2,θνi,(x′,y′).

149


Cρ/2,θνi,z

ρ/4

Ai

V

Di

ρ/2

θ/2

θ

K

z

νi

Figure 4.2: Illustration of the proof of Theorem 5.8.

Then Ai ∩ V ⊂ ∂Di ∩ V : indeed if (x′, y′) ∈ Ai ∩ V , then (x′, y′) ∈ Di ∩ V , and(x′, y′) can not lie in the interior of Di, otherwise for λ > 0 small enough, we wouldhave (x′, y′) − λν ∈ Di, which would imply that (x′, y′) lies in the interior of oneof the cones forming Di, and therefore in the interior of K, which is absurd since(x′, y′) ∈ ∂K.

Step 3. The set ∂Di ∩ V is a Lipschitz graph of constant√

(2θ/ρ)2 − 1. Moreprecisely let us prove that ∂Di ∩ V is equal to

Gi =(x′, y′) : x′ ∈ BN−1(x, ρ/4)

and y′ = maxy′′; (x′, y′′) ∈ ∂C for one of the cones C forming Di.

First of all, it is easy to show that Di is closed, and that the maximum in thedefinition of Gi exists and is not equal to y + θ

2; otherwise there would exist a

cone C in Di such that (x, y) ∈ int(C) ⊂ int(K), which is absurd. The inclusionGi ⊂ ∂Di∩V follows from the same argument used for the inclusion Ai∩V ⊂ ∂Di∩Vin Step 2. Conversely, let us fix (x′, y′) ∈ ∂Di ∩ V . Then (x′, y′) ∈ Di since Di isclosed, so that (x′, y′) belongs to the trace on V of one of the cones forming D, letus say (x′, y′) ∈ C. But then (x′, y′) can not belong to int(C), otherwise we wouldhave (x′, y′) ∈ int(Di), so we deduce that (x′, y′) ∈ ∂C ∩ V . Moreover if there existsy′′ > y′ such that (x′, y′′) ∈ ∂C′ for some other of the cones C′ forming Di, then we

150


must have (x′, y′) ∈ int(C′) ∩ V ⊂ int(Di), which is absurd, and proves that y′ isequal to the maximum in the definition of Gi, and that ∂Di ∩ V ⊂ Gi. Therefore∂Di ∩ V is a Lipschitz graph of constant µ =

√(2θ/ρ)2 − 1 as the supremum of

graphs of cones of same parameters ρ and θ.

Step 4. Estimate of the perimeter of Ai in V. It follows from Step 3 that ∂Di ∩ V isHN−1 measurable with

HN−1(∂Di ∩ V ) ≤ LN−1(BN−1(x, ρ/4))√

1 + µ2,

whence

HN−1(Ai ∩ V ) ≤ ωN−1

(ρ4

)N−1 2θ

ρ,

where ωj denotes the volume of the unit ball of Rj .

Step 5. Covering of Ai with balls of fixed radius. By Besicovitch’s covering theorem(see [38]), there exists a constant ξN depending only on N such that for any ε > 0and R > 0, there exist numbers Γ1, . . . ,ΓξN

and a finite family (xkj) (for 1 ≤ k ≤ ξNand 1 ≤ j ≤ Γk) of points of Ai ∩ B(0, R) such that

Ai ∩ B(0, R) ⊂ξN⋃

k=1

Γk⋃

j=1

B(xkj, ε),

for each k, the balls B(xkj, ε), 1 ≤ j ≤ Γk, are pairwise disjoint.

The family (xkj)j is a priori only countable, but has to be finite by boundednessof Ai and because the radius of covering balls is fixed. We now want to estimate∑ξN

k=1 Γk. Let us therefore compute

∫

K∩B(0,R+ε)

ξN∑

k=1

Γk∑

j=1

1B(xkj ,ε).

On the one hand, we have

ξN∑

k=1

∫

K∩B(0,R+ε)

Γk∑

j=1

1B(xkj ,ε) ≤ ξNLN(K ∩ B(0, R + ε)), (5.7)

because for each k, the balls B(xkj , ε) are pairwise disjoint. On the other hand,for each k and j, the ball B(xkj, ε) contains a fixed portion of the cone Cρ/2,θ

νi,xkj ,portion which is included in K ∩ B(0, R + ε) by the interior cone property, sincexkj ∈ Ai ∩ B(0, R). We call

γ := LN(B(xkj, ε) ∩ Cρ/2,θνi,xkj

)

the volume of this portion of cone, the computation of which is done in Step 7. Notethat γ is independent of xkj . Therefore

∫

K∩B(0,R+ε)

ξN∑

k=1

Γk∑

j=1

1B(xkj ,ε) =

ξN∑

k=1

Γk∑

j=1

∫

K∩B(0,R+ε)

1B(xkj ,ε) ≥ γ

ξN∑

k=1

Γk. (5.8)

151


From (5.7) and (5.8), we deduce that

ξN∑

k=1

Γk ≤ ξNγLN(K ∩ B(0, R + ε)).

Since B((x, y), ε) ⊂ V = BN−1(x, ρ/4) × (y − θ/2, y + θ/2) , as soon as

ε < minρ/4, θ/2 = ρ/4,

we see that Ai ∩ B(0, R) can be covered by∑ξN

k=1 Γk cylinders of the form of Vcentered at points of Ai ∩ B(0, R), so that, from Step 4,

HN−1(Ai ∩ B(0, R)) ≤ξN∑

k=1

Γk ωN−1

(ρ4

)N−1 2θ

ρ

≤ ξNγωN−1

(ρ4

)N−1 2θ

ρLN(K ∩ B(0, R + ε)).

Step 6. Sum for all axes. What we have done does not depend on the fixed directionaxis νi and we know, thanks to Step 1, that ∂K is the union of less than p = β(N)

(ρ/2θ)N−1

sets of the form Ai, so that we finally have

HN−1(∂K ∩ B(0, R)) ≤ β(N)

(ρ/2θ)N−1

ξNγωN−1

(ρ4

)N−1 2θ

ρLN(K ∩ B(0, R+ ε)),

which gives (5.6).

Step 7. Computation of the value of γ. As soon as ε ≤√θ2 − (ρ/2)2) (which

corresponds to the length of the longest segment included in ∂Cρ/2,θνi,xkj

), then B(xkj , ε)

contains at least the straight portion of Cρ/2,θνi,xkj

of length

l =µ√

1 + µ2ε = µ

ρ

2θε,

the volume of which equals

ωN−1

N

lN

µN−1=ωN−1

Nµ( ρ

2θε)N

.

This gives a lower bound for γ. Moreover, we obtain a more precise estimate for Λin (5.6): since ρ < θ, we see that ρ/4 ≤

√θ2 − (ρ/2)2, so that sending ε to ρ/4, we

get

HN−1(∂K ∩ B(0, R)) ≤ 4N+1Nβ(N)ξN1

ρ

(θ/ρ)2N

√(2θ/ρ)2 − 1

LN(K ∩ B(0, R + ρ/4)).

152


5.4 Propagation of the interior cone property

We want to prove that the interior cone property is preserved for sets whose evo-lution is governed by the eikonal equation (1.3). We make the following assumptionon c:

(H4) The function c(·, t) is Lipschitz continuous with a constant independent oft ∈ [0, T ] and, for all R > 0, there exists an increasing modulus of continuity ωR

such that, for all x ∈ B(0, R), t, s ∈ [0, T ], then

|c(x, t) − c(x, s)| ≤ ωR(|t− s|).

Theorem 5.9. Assume that c satisfies (H1) and (H4) and that u0 satisfies (H2)and (H3). Let u be the unique uniformly continuous viscosity solution of (1.3).Then there exist ρ > 0 and θ > 0 depending only on K0, c, c, C and ωR withR = R0 + cT , such that K(t) = x ∈ RN ; u(x, t) ≥ 0 has the interior coneproperty of parameters ρ and θ for all t ∈ [0, T ]. More precisely, let r > 0 be suchthat K0 has the interior ball property of radius r > 0, then we can choose

θ = min

c2

6Cc, c ω−1

R (c/4) , r

and ρ =

c

2cθ.

Proof.Step 1. Minimal time function. We first claim that t 7→ u(x, t) is nondecreasingfor any x ∈ RN and, if u(x, t) = 0, then u(x, s) > 0 for any s ∈ (t, T ]. Indeed, let(x, t) ∈ RN × [0, T ] such that u(x, t) = 0. Since u is Lipschitz continuous and thelower-gradient bound estimate (5.2) holds almost everywhere in |u| < η/2, thereexists r0 > 0 small enough such that, for all 0 ≤ t ≤ s ≤ t+ r0 and r ≤ r0,

∫

B(x,r)

(u(y, s)− u(y, t))dy =

∫ s

t

∫

B(x,r)

ut(y, τ)dydτ

=

∫ s

t

∫

B(x,r)

c(y, τ)|Du(y, τ)|dydτ

≥ LN(B(x, r))cη(s− t).

Dividing by LN(B(x, r)) and letting r → 0, we get that u(x, s) > 0 if t < s ≤ t+ r0.The proof that t 7→ u(x, t) is nondecreasing for any x ∈ R

N can be obtained in asimilar way by simpler arguments (in particular we do not need (5.2)).

Therefore, the minimal time function

v(x) = mint ∈ [0, T ]; u(x, t) ≥ 0

is defined at points x ∈ K(T ), and for any t ∈ [0, T ],

x ∈ RN ; u(x, t) ≥ 0 = x ∈ R

N ; v(x) ≤ t,x ∈ R

N ; u(x, t) = 0 = x ∈ RN ; v(x) = t.

153


Moreover, v is 1/c -Lipschitz in K(T ): let us fix x and y in K(T ) with v(x) ≤ v(y).The function

u : (z, t) 7→ sup|z′−z|≤c |t−v(x)|

u(z′, v(x))

is the unique uniformly continuous viscosity solution (see [14]) of

ut(z, t) = c|Du(z, t)| in RN × (v(x), T ),

u(·, v(x)) = u(·, v(x)) in RN .

The comparison principle for continuous viscosity solutions implies that u ≤ u inRN × [v(x), T ]. In particular

u(y,1

c|x− y| + v(x)) ≤ u(y,

1

c|x− y| + v(x)),

which implies by definition of u and v that

0 = u(x, v(x)) ≤ u(y,1

c|x− y| + v(x)) ≤ u(y,

1

c|x− y| + v(x)).

The Lipschitz property of v follows, since we deduce that

v(y) ≤ 1

c|x− y| + v(x).

Step 2. Interior cone property at time t ∈ [µ, T ] for some µ > 0. To prove the claimof the theorem, we will use arguments from control theory. To do this we need thevelocity c to be C1 in space, an additional condition that we can assume withoutloss of generality, replacing c by a suitable space convolution cδ of c. Then we provethe result for cδ, and, letting δ → 0+, obtain the desired result since the constantsθ and ρ do not depend on δ.

It is well-known that, for each time t, the set K(t) can be seen as the reachableset from K0 for the controlled system

x′(t) = c(x(t), t)a(t) for t ∈ [0, T ], (5.9)

where the control a takes its values in the unit closed ball. Let x be an extremaltrajectory, i.e. a trajectory satisfying x(T ) ∈ ∂K(T ). For such a trajectory, it is easyto see that t 7→ u(x(t), t) is non-decreasing, from which we infer that x(t) ∈ ∂K(t)for any t ∈ [0, T ], that is to say, v(x(t)) = t.

The Pontryagine maximum principle (see [35]) implies the existence of an adjointp such that the following system is satisfied on [0, T ]:

x′(t) = c(x(t), t)

p(t)

|p(t)| ,

−p′(t) = Dc(x(t), t)|p(t)|.(5.10)

154


From now on, we fix 0 ≤ t ≤ T. From (5.10) and the regularity of c we infer that, ifwe set M = 3Cc, then for any s ∈ [0, t ],

|x′(s) − x′(t )| ≤ M(t− s) + ωR(t− s),

where R := R0 +cT is given by Lemma 5.3. By integration on [t, t ], we deduce that,for any t ∈ [0, t ],

|x(t ) − x(t) − x′(t )(t− t)| ≤ M

2(t− t)2 + ωR(t− t)(t− t). (5.11)

Let x ∈ ∂K(t ), and let x(·) be an extremal trajectory with x(t ) = x. We are going toshow that for any t ∈ [0, t ], the ball B(t) of radius r(t) centered at x(t )−x′(t )(t−t) iscontained inK(t ) for some r(t) to determine, i.e. that we have for any ξ ∈ B(0, r(t)),

v (x(t ) − x′(t )(t− t) + ξ) ≤ t.

We therefore estimate, using the Lipschitz continuity of v and (5.11),

t− v (x(t ) − x′(t )(t− t) + ξ)

≥ t− v (x(t ) − x′(t )(t− t)) − 1

c|ξ|

≥ t− v(x(t)) − 1

c

(M

2(t− t)2 + ωR(t− t)(t− t)

)− 1

cr(t)

= t− t− 1

c

(M

2(t− t)2 + ωR(t− t)(t− t) + r(t)

).

Thus if we set r(t) = 12c(t− t), the above quantity is nonnegative as soon as

t− t ≤ c

2Mand ωR(t− t) ≤ 1

4c.

For this choice, it follows

B(t) = B (x(t ) − x′(t )(t− t), r(t))

=

x(t ) − x′(t )

|x′(t )| |x′(t )|(t− t) +

c

2|x′(t )| |x′(t )|(t− t) ξ, ξ ∈ B(0, 1)

⊂ K(t ).

Since x(t ) = x and c ≤ |x′(t )| ≤ c, this proves the interior cone property at x assoon as t ≥ µ := min

(c

2M, ω−1

R (c/4)), of parameters

ρ1 =c

2cθ1, with θ1 = min

(c2

2M, cω−1

R (c/4)

).

Step 3. Interior cone property for small times t ∈ [0, µ]. With the previous notation,let x ∈ ∂K(t ) and x(·) be an extremal trajectory of (5.9) with x(t ) = x. Let us

155


recall that the regularity of K0 implies that it has the interior ball property, i.e.there exists r > 0 independent of y ∈ ∂K0 such that

B(y − ν(y)r, r) ⊂ K0,

where ν(y) is the unit outer normal to K0 at y ∈ ∂K0. Note that, as a consequence,K0 has the interior cone property at x(0) of parameters ρ = r/2 and θ = r, withaxis ν(x(0)). We see by the regularity of K0 that ν(x(0)) = p(0)/|p(0)|, so that

B(x(0) − p(0)

|p(0)|r, r) ⊂ K0. (5.12)

We will prove that, for t ≤ µ, K(t ) has the interior cone property of parametersρ = r/2 and θ = r. Let y ∈ Cr/2,r

ν,x with ν = − p(t )|p(t )| . We write y as

y = x− p(t )

|p(t )|λ+1

2λξ, (5.13)

where 0 ≤ λ ≤ r and |ξ| ≤ 1. Let y(·) be the solution of

y′(t) = c(y(t), t)

p(t)

|p(t)| for t ∈ [0, t ],

y(t ) = y,

where p(·) is the adjoint associated with x(·) by (5.10). It is enough to prove thaty(0) ∈ K0, since then y = y(t ) ∈ K(t ). Because of (5.12), we only have to showthat

∣∣∣∣y(0) −(x(0) − p(0)

|p(0)|λ)∣∣∣∣ ≤ λ.

Moreover, we remark that (5.13) implies that

∣∣∣∣y(t ) −(x(t ) − p(t )

|p(t )|λ)∣∣∣∣ =

∣∣∣∣1

2λξ

∣∣∣∣ ≤λ

2.

Let us therefore set

f(t) =

∣∣∣∣y(t) − x(t) + λp(t)

|p(t)|

∣∣∣∣2

,

so that f(t ) ≤ λ2

4. It only remains to prove that f(0) ≤ λ2. But, differentiating f ,

156


we find

f ′(t) = 2 〈y(t) − x(t), y′(t) − x′(t)〉 + 2λ

⟨y′(t) − x′(t),

p(t)

|p(t)|

⟩

+2λ

⟨y(t) − x(t),

d

dt

p(t)

|p(t)|

⟩

= 2

⟨y(t) − x(t), (c(y(t), t) − c(x(t), t))

p(t)

|p(t)|

⟩

+2λ

⟨(c(y(t), t) − c(x(t), t))

p(t)

|p(t)| ,p(t)

|p(t)|

⟩

+2λ

⟨y(t) − x(t),

p′(t)

|p(t)| −p(t) 〈p(t), p′(t)〉

|p(t)|3⟩

≥ −2C|y(t) − x(t)|2 − 2λC|y(t) − x(t)| − 2λ|y(t) − x(t)|∣∣∣∣p′(t)

|p(t)|

∣∣∣∣

−2λ|y(t) − x(t)|∣∣∣∣p(t) 〈p(t), p′(t)〉

|p(t)|3∣∣∣∣ .

Thanks to (5.10), we know that∣∣∣∣p′(t)

|p(t)|

∣∣∣∣ ≤ C and

∣∣∣∣p(t)〈p(t), p′(t)〉

|p(t)|3∣∣∣∣ ≤ C,

so thatf ′(t) ≥ −2C|y(t) − x(t)|2 − 6λC|y(t) − x(t)|.

But if we set g(t) = |y(t) − x(t)|2, then

g′(t) = 2〈y(t) − x(t), y′(t) − x′(t)〉 ≥ −2C|y(t) − x(t)|2 = −2Cg(t),

which implies that for all t ∈ [0, t ]

g(t)e2Ct ≤ g(t )e2Ct,

that is to say thanks to (5.13)

|y(t) − x(t)| ≤ |y − x|eC(t−t) ≤ 3λ

2eCt.

We therefore obtain

f ′(t) ≥ −2C

(3λ

2eCt

)2

− 6λC3λ

2eCt = −

(9

2Ce2Ct + 9CeCt

)λ2.

If we set k = 92Ce2Ct + 9CeCt, we finally have

f(0) ≤ f(t ) + kλ2t ≤ λ2

4+ kλ2t ≤ λ2

157


as soon as kt ≤ 34. Thus if we set b to be the unique solution of 9

2be2b + 9beb = 3

4

(b > 0), we get that f(0) ≤ λ2 as soon as t ≤ b/C. If we assume that

b

C≥ c

2M=

c

6Cc,

which is always possible by reducing c or increasing c, we see that K(t ) has theinterior cone property of parameters ρ2 = r/2 and θ2 = r for all 0 ≤ t ≤ µ (notethat the parameters ρ2, θ2 depend only on K0).

Step 4. End of the proof. We remark that

ρ1

θ1=

c

2c≤ 1

2=ρ2

θ2,

whence we finally obtain that for any t ≥ 0, K(t ) has the interior cone property ofparameters ρ = c

2cθ and θ = minθ1, θ2.

Aknowledgment. This work was partially supported by the ANR (Agence Na-tionale de la Recherche) through MICA project (ANR-06-BLAN-0082).

158

Chapitre 5

Convergence of approximation

schemes for nonlocal front

propagation equations

Ce chapitre est issu de l’article [59].

On donne un résultat de convergence de schémas numériques pour des équationsde propagations de fronts non-locales générales. Ces schémas sont basés sur lanotion de solution faible décrite dans l’introduction et dont l’existence a été étudiéeau chapitre 2. On donne aussi des exemples de tels schémas dans le cas des deuxéquations modèle, la dynamique des dislocations et le système de FitzHugh-Nagumo.

Chapitre 5. Approximation schemes for nonlocal equations

Abstract

We provide a convergence result for numerical schemes approximating

nonlocal front propagation equations. Our schemes are based on a re-

cently investigated notion of weak solution for these equations. We also

give examples of such schemes, for a dislocation dynamics equation, and

for a FitzHugh-Nagumo type system.

Key words and phrases: Approximation schemes, front prop-agations, level-set approach, nonlocal Hamilton-Jacobi equations,second-order equations, viscosity solutions, L1 dependence in time,dislocation dynamics, FitzHugh-Nagumo system.

1 Introduction

We are concerned with numerical approximation for nonlocal equations of theform

ut(x, t) = H [1u≥0](x, t,Du,D

2u) in RN × (0, T ),u(·, 0) = u0 in RN ,

(1.1)

which, in the level-set approach for front propagation, describe the movement of afamily K(t)t∈[0,T ] of compact subsets of R

N such that

K(t) = x ∈ RN ; u(x, t) ≥ 0

for some function u : RN × [0, T ] → R. Here ut, Du and D2u denote respectivelythe time derivative, space gradient and space Hessian matrix of u, while 1A denotesthe indicator function of any set A.

The function H corresponds to the velocity of the front. In our setting, itdepends not only on local properties of the front, such as its position, the time,the normal direction and its curvature matrix, but also, at time t, on the familyK(s)s∈[0,t] itself. This nonlocal dependence is carried by the notation H [1u≥0]:for any indicator function χ or more generally for any χ ∈ L∞(RN × [0, T ]) withvalues in [0, 1], the Hamiltonian H [χ] depends on χ in a nonlocal way; typically inour examples, it is obtained by a convolution procedure between χ and a physicalkernel (either only in space or in space and time). In particular, H [χ] is continuousin space but has no particular regularity in time. However, the H [χ]−equation isalways well posed.

More precisely, we assume that for any χ ∈ L∞(RN × [0, T ]; [0, 1]) with boundedsupport, H [χ](x, t, p, A) defines a measurable function of (x, t, p, A) ∈ RN × [0, T ]×RN \0×SN , while for almost every t ∈ [0, T ], H [χ](x, t, p, A) defines a continuousfunction of (x, p, A). Here SN denotes the set of real square symmetric matrices ofsize N .

Let us specify the class of equations that we consider: first of all, we are in-terested in front propagation equations, and therefore assume that for any χ ∈L∞(RN × [0, T ]; [0, 1]) with bounded support, the equation ut = H [χ](x, t,Du,D2u)

160

1. Introduction

is geometric, and that the upper and lower semicontinuous envelopes of the Hamil-tonian H [χ] with respect to (x, p, A) satisfy, for any x ∈ RN and almost all t ∈ [0, T ],

H [χ]∗(x, t, 0, 0) = H [χ]∗(x, t, 0, 0) = 0. (1.2)

We also assume that this equation is degenerate parabolic, which means that forany (x, p) ∈ RN × RN \ 0, for almost every t ∈ [0, T ] and for all A,B ∈ SN , wehave

H [χ](x, t, p, A) ≤ H [χ](x, t, p, B) if A ≤ B,

where ≤ stands for the usual partial ordering for symmetric matrices.

The initial datum u0 : RN → R is a bounded and Lipschitz continuous function

on RN which represents the initial front, i.e. such that

u0 ≥ 0 = K0 and u0 = 0 = ∂K0

for some fixed compact set K0 ⊂ RN . Since in the level-set approach, the familyK(t)t∈[0,T ] only depends on the 0-level set of u0 (see [46]), we assume for simplicitythat there exists R0 > 0 such that

u0(x) = −1 if |x| ≥ R0, (1.3)

where | · | denotes the standard Euclidean norm on RN . For computational reasons,

we ask the equation to preserve this property of compactness of the front. Essentially,this means that there exists a continuous function R on [0, T ] such that R(0) = R0

and the solution of ut = H [χ](x, t,Du,D2u) with initial datum u0 has the followingproperty:

χ(x, t) = 0 for a.e. (x, t) s.t. |x| ≥ R(t) ⇒ u(x, t) = −1 for any (x, t) s.t. |x| ≥ R(t).

Finally, for the same computational reasons, we point out that even though existenceof solutions to (1.1) is known in a more general setting (see [17]), in this chapterwe consider equations depending on the past, which means that H [χ](x, t, p, A) onlydepends on χ(·, s) for 0 ≤ s ≤ t.

The main issue linked with these nonlocal equations is the fact that they donot satisfy a comparison principle (or, geometrically, an inclusion principle on thefronts). Indeed, in general the fact that u1 ≥ 0 ⊂ u2 ≥ 0 does not imply thatH [1u1≥0] ≤ H [1u2≥0]. A consequence of this absence of monotonicity is thatone cannot build viscosity solutions to (1.1) by the classical methods, a comparisonprinciple being crucial for both existence and uniqueness of a solution.

To overcome these difficulties, a notion of weak solution to (1.1) has thereforebeen introduced in [16, 17]. It uses the notion of L1-viscosity solution, a notion ofsolution adapted to Hamiltonians H [χ] which are merely measurable in time. Werefer to the general introduction for the definition of L1-viscosity solutions and to[50, 61, 62, 27, 26] for a complete presentation of the theory.

Let us now recall the definition of a weak solution to (1.1):

161


Definition 1.1. Let u : RN × [0, T ] → R be a continuous function. We say that u

is a weak solution of (1.1) if there exists χ ∈ L∞(RN × [0, T ]; [0, 1]) such that:

1. u is a L1-viscosity solution of

ut(x, t) = H [χ](x, t,Du,D2u) in RN × (0, T ),u(·, 0) = u0 in RN .

(1.4)


1u(·,t)>0 ≤ χ(·, t) ≤ 1u(·,t)≥0 a.e. in RN . (1.5)


1u(·,t)>0 = 1u(·,t)≥0 a.e. in RN .

In Chapter 2 or [17], with Barles, Cardaliaguet and Ley, we proved a generalexistence result for these nonlocal equations. In the framework described above,the essential assumptions under which existence is known are the following; theyconcern the local equation (1.4), where the nonlocal dependence is frozen, that is tosay, 1u≥0 is replaced by a fixed function χ ∈ L∞(RN × [0, T ]; [0, 1]):

(A1) If χn χ weakly-∗ in L∞(RN × [0, T ]; [0, 1]), and if Supp(χn) is uniformlybounded, then for all (x, t, p, A) ∈ RN × [0, T ] × RN \ 0 × SN ,

∫ t

0

H [χn](x, s, p, A)ds −→n→+∞

∫ t

0


locally uniformly in x, t, p, A.

(A2) A comparison principle holds for (1.4): for any fixed χ ∈ L∞(RN ×[0, T ]; [0, 1])with bounded support, if u is a bounded and upper semicontinuous L1-viscosity sub-solution of (1.4) and v is a bounded, lower semicontinuous L1-viscosity supersolutionof (1.4), then u ≤ v in RN × [0, T ).

These assumptions are the classical ingredients to carry out a stability argument:assumption (A1) provides stability for L1-viscosity solutions under very weak con-vergence of the Hamiltonians, thanks to a new stability result of Barles [15], whileassumption (A2) enables us to identify the limit by a comparison principle. Thisis the idea of the proof of the existence result of Chapter 2 or [17]. We assumethroughout the chapter that these assumptions hold, and we refer to [26, 62] forconditions on H [χ] under which (A2) holds.

We also point out that assumption (A2) implies that for any fixed χ ∈ L∞(RN ×[0, T ]; [0, 1]) with bounded support, (1.4) has a unique continuous L1-viscosity so-lution u : RN × [0, T ] → R. Combined with (1.2), which shows that constants aresolutions of (1.4), it also implies the existence of uniform bounds on the solutionsof (1.4), independent of χ.

162

2. Convergence of approximation schemes

Considering this existence result, our motivation is to provide numerical schemes,and a general convergence result, for these nonlocal and nonmonotone front propa-gation equations with L1 dependence in time. This work is inspired by [23] whereBarles and Souganidis proved a general convergence result for monotone, stable andconsistent schemes in the local framework. We also refer to the works of Cardaliaguetand Pasquignon [32] and Slepčev [70] on the approximation of moving fronts in thenonlocal but monotone case.

This chapter is organized as follows: in Section 2, we define a class of approx-imation schemes and prove the general convergence result. In Section 3, we givetwo explicit examples of such schemes, for the dislocation dynamics equation andFitzHugh-Nagumo system already often mentioned.

Notation. In what follows, | · | denotes the standard Euclidean norm on RN orSN , B(x,R) (resp. B(x,R)) is the open (resp. closed) ball of radius R centered atx ∈ RN . We denote the essential supremum of f ∈ L∞(RN) with values in R, RN

or SN , f ∈ L∞(R; R) or f ∈ L∞(RN × [0, T ]; R), by ‖f‖∞.

2 Convergence of approximation schemes

Let h = T/n for some n ∈ N∗, and ∆1, . . . ,∆N ∈ (0, 1) be our respective timeand space steps: a choice of h determines fixed ∆i’s by the relation ∆i = λi h forλi > 0 fixed. We define for (i1, . . . , iN) ∈ ZN , xi1,...,iN = (i1∆1, . . . , iN∆N), and

Qi1,...,iN =

N∏

k=1

[(ik − 1/2)∆k, (ik + 1/2)∆k).

Let us also define the space grid

Πh =⋃

(i1,...,iN )∈ZN

xi1,...,iN,

and for x = (x1, . . . , xN ) ∈ RN , its projection on this grid,

xh := ([x1/∆1 + 1/2]∆1, . . . , [xN/∆N + 1/2]∆N) ∈ Πh,

where [·] denotes the integer part, so that if x ∈ Qi1,...,iN , then xh = xi1,...,iN .

For x ∈ Πh, k ∈ N such that kh ≤ T , u : Πh → R and χ : Πh × [0, T ] → [0, 1]with bounded support, we define an approximate Hamiltonian Hh[χ](x, kh, u) whichdepends on

χ(xi1,...,iN , lh)(i1,...,iN )∈ZN , 0≤l≤k and u(xi1,...,iN )(i1,...,iN )∈ZN .

We keep in mind thatHh[χ](x, kh, u) possibly depends on the entire history χ(·, lh)for l up to k.

163


We consider approximation schemes of the following form: for any k ∈ N suchthat (k + 1)h ≤ T , and for any x ∈ Πh, we set


(2.1)

We finally extend uh to a piecewise constant function on RN × [0, T ] by setting forany (x, t),

uh(x, t) = uh(xh, [t/h]h).

In particular we have for any x ∈ RN ,

uh(x, 0) = u0(xh).

Let us now state our assumptions on Hh; in what follows C2b (RN ; R) denotes the

set of C2 functions on RN such that the norm

‖φ‖ = ‖φ‖∞+‖Dφ‖∞+‖D2φ‖∞ = supx∈RN

|φ(x)|+ supx∈RN

|Dφ(x)|+ supx∈RN

|D2φ(x)| (2.2)

is finite. Let us first state an assumption on the behavior of Hh with respect to itslast variable, which represents space derivatives. It is a trivial assumption whichis linked to the fact that the equation ut = H [χ](x, t,Du,D2u) is geometric forany fixed χ; it will be satisfied for all reasonable schemes at no cost, so we state itseparately:

(H0) Consistency with respect to derivatives:(i) For any x ∈ Πh, k, h with kh ≤ T , u : Πh → R, λ ∈ R, and any function

χ : Πh × [0, T ] → [0, 1] with bounded support,

Hh[χ](x, kh, u+ λ) = Hh[χ](x, kh, u), and Hh[χ](x, kh, 0) = 0.

(ii) There exists r ∈ N∗ such that for any x ∈ Πh, k, h with kh ≤ T , for anyχ : Πh × [0, T ] → [0, 1] with bounded support, and for all u, v : Πh → R,

if u(y) = v(y) ∀ y ∈ Πh s.t. ∀i, |xi−yi| ≤ r∆i, then Hh[χ](x, kh, u) = Hh[χ](x, kh, v).

We easily deduce from this and (1.3) that there exists R = R0 + rT√N maxλi

such that if uh is defined by the scheme (2.1), then uh(x, t) = −1 if x ∈ RN \B(0, R),

for all t ∈ [0, T ]; hence we only need to consider functions χ with uniformly boundedsupport. This shows in addition that the domain of space computation is uniformlybounded. In particular we set Bh(R

N × [0, T ]; [0, 1]) to be the set of functions χdefined on RN × [0, T ] with values in [0, 1] such that Supp(χ) ⊂ B(0, R)× [0, T ] andχ is constant on each of the sets ∪ Qi1,...,iN × [kh, (k + 1)h).

Our assumptions are the following:

(H1) Hh is conditionally monotone: for any x ∈ Πh, k, h with kh ≤ T , for anyχ ∈ Bh(R

N × [0, T ]; [0, 1]), and for all u, v : Πh → R,

u ≤ v ⇒ u(x) + hHh[χ](x, kh, u) ≤ v(x) + hHh[χ](x, kh, v).

164


(H2) Hh is stable: there exists L > 0 such that for any x ∈ Πh, k, h with kh ≤ T ,and χ ∈ Bh(R

N × [0, T ]; [0, 1]), the solution uh of (2.1) satisfies

|uh(x, kh)| ≤ L.

(H3) Hh is consistent with H : for any x ∈ RN and φ ∈ C2b (R

N ; R) such thatDφ(x) 6= 0, if χh ∈ Bh(R

N × [0, T ]; [0, 1]) is such that χh χ weakly-∗ in L∞(RN ×[0, T ]; [0, 1]) as h→ 0, then

h

[t/h]−1∑

l=0

Hh[χh](xh, lh, φ) −→h→0

∫ t

0

H [χ](x, s,Dφ(x), D2φ(x)) ds

locally uniformly for t ∈ [0, T ] (the sum is set to 0 if t < h).

(H4) Regularity : for any compact subset K of RN×C2b (RN ; R), there exist uniformly

bounded moduli of continuity mh such that for any h > 0, (x, φ), (y, ψ) ∈ K withx, y ∈ Πh, for any k, h with kh ≤ T , and any χ ∈ Bh(R

N × [0, T ]; [0, 1]),

|Hh[χ](x, kh, φ)−Hh[χ](y, kh, ψ)| ≤ mh(|x−y|+|Dφ(x)−Dψ(y)|+|D2φ(x)−D2ψ(y)|),

and such that mh(η) → 0 as h, η → 0.

Assumptions (H1) to (H3) are the classical assumptions introduced by Barlesand Souganidis in [23]. Moreover (H3) is the discrete equivalent of (A1) on the weakconvergence of the Hamiltonians. As a matter of fact, the proof of our convergencetheorem is based on the proof of the stability result of [15], the key assumption ofwhich is (A1). Finally assumption (H4) appears naturally alongside (H3), just asin the continuous case (see [15]).

Remark 2.1. Under assumption (H0) (ii), if (H1) holds, then it also holds for allfunctions u and v such that u(y) ≤ v(y) for any y ∈ Πh with |xi − yi| ≤ r∆i for alli = 1 . . .N , that is, also for functions that are comparable only locally. Indeed in thiscase, we can change u and v to 0 out of the set y ∈ Πh; |xi−yi| ≤ r∆i ∀i = 1 . . .N.This provides new functions u and v such that u ≤ v in Πh, whence, using (H1),

u(x) + hHh[χ](x, kh, u) ≤ v(x) + hHh[χ](x, kh, v).

But u(x) = u(x), Hh[χ](x, kh, u) = Hh[χ](x, kh, u) thanks to (H0) (ii), and thesame holds for v. This proves our assertion.

In the same spirit, we notice that assumption (H4) also holds for two functions φand ψ in C2(RN ; R), because one can always modify φ and ψ to obtain new functionsin C2

b (RN ; R) without changing the values of Hh[χ](x, kh, φ) or Hh[χ](y, kh, ψ).

Let us now state our main result:

Theorem 2.2. Assume that assumption (A2) holds. Let u0 be a bounded and Lip-schitz continuous function which satisfies (1.3). Let (uh)h be defined by the scheme(2.1) satisfying assumptions (H0) to (H4).

165


Then there exist hn → 0, u ∈ C0(RN × [0, T ]; R) and χ ∈ L∞(RN × [0, T ]; [0, 1])such that uhn → u locally uniformly in RN × [0, T ], 1uhn≥0 χ weakly-∗ inL∞(RN × [0, T ]; [0, 1]) and (u, χ) satisfies (1.4).

Moreover, any such (u, χ) satisfies (1.5), so that u is a weak solution of (1.1).If in addition (1.1) has a unique weak solution u, then the whole sequence (uh)converges locally uniformly to u in RN × [0, T ].

Proof. By compactness of L∞(RN × [0, T ]; [0, 1]) for the weak-∗ topology, we canfind χ ∈ L∞(RN × [0, T ]; [0, 1]) and (hn) converging to 0 such that

1uhn≥0 χ weakly- ∗ in L∞(RN × [0, T ]; [0, 1]).

By the stability assumption (H2), there exists L > 0 such that ‖uh‖∞ ≤ L for anyh. We can therefore set

u(x, t) = lim sup∗(uhn)(x, t)

= lim supuh′n(xn, knh

′n); (h′n) ⊂ (hn), xn → x with xn ∈ Πhn,

knh′n → t with kn → +∞,

which defines a bounded upper semi-continuous function on RN × [0, T ]. Let usprove that u is a L1-viscosity subsolution of (1.4). We could prove in the same waythat u(x, t) = lim infuh′

n(xn, knh

′n); (h′n) ⊂ (hn), xn → x, knh

′n → t is a bounded

L1-viscosity supersolution of (1.4).

Step 1. We first prove that for any x ∈ RN , u(x, 0) ≤ u0(x). To do this we adapt

the proof of the same statement in the proof of Theorem 4.1 of Chapter 2 or [17].First of all, u0 is Lipschitz continuous, so that for any fixed 0 < ε ≤ 1, we have, forany x, y ∈ RN ,

u0(y) ≤ u0(x) + ‖Du0‖∞|x− y| ≤ u0(x) +|x− y|2

2ε2+

‖Du0‖∞ε2

2.

We fix x and set φ(y) = |x−y|2/2ε2. Using the above inequality, the function definedby

ψε(y, khn) = u0(x) + φ(y) +‖Du0‖∞ε2

2+ Cε khn

satisfies uhn(y, 0) = u0(y) ≤ ψε(y, 0) for all y ∈ Πh. Moreover, using (H0) (i),we see that ψε is a supersolution of (2.1) associated to Hhn[1uhn≥0] in the ballB(x, ε+ rT

√N maxλi), provided that Cε is large enough, namely as soon as

Hhn [1uhn≥0](y, khn, φ) ≤ Cε for all y with |x−y| < ε+rT√N maxλi and khn ≤ T.

This condition can be fulfilled using (H4) and the fact thatHhn[1uhn≥0](y, khn, 0) =0 (assumption (H0) (i)). Indeed, for some uniformly bounded moduli of continuity,we have

Hhn[1uhn≥0](y, khn, φ) ≤ mhn(|Dφ(y)|+ |D2φ(y)|)

166


for any n ∈ N, y ∈ Πhn such that |x − y| < ε + rT√N maxλi, and khn ≤ T . The

function φ does not belong to C2b (R

N ; R), but using Remark 2.1, we recall that (H4)can also be applied to two functions in C2(RN ; R). By the conditional monotonicityassumption (H1) (using again Remark 2.1), we obtain that for any y ∈ Πhn with|y − x| < ε+ r(T − hn)

√N max λi,

uhn(y, hn) ≤ ψε(y, hn).

Reproducing the argument, we get that for any y ∈ Πhn with |y − x| < ε and k, hn

with khn ≤ T ,uhn(y, khn) ≤ ψε(y, khn),

and in particular

u(x, 0) ≤ lim sup∗ψε(x, 0) = u0(x) +‖Du0‖∞ε2

2.

Sending ε to 0 proves the claim.

Step 2. Now let φ ∈ C2(RN × (0, T ); R) and b ∈ L1((0, T ); R) be such that

(x, t) 7→ u(x, t) − φ(x, t) −∫ t

0

b(s) ds

has a local maximum at some (x0, t0) ∈ RN × (0, T ). Let G be a continuous functionsuch that for almost all t in a neighborhood of t0, for all (x, p, A) in a neighborhoodof (x0, Dφ(x0, t0), D

2φ(x0, t0)),

H [χ]∗(x, t, p, A) − b(t) ≤ G(x, t, p, A).

To check the L1-viscosity subsolution property, we have to prove that

φt(x0, t0) ≤ G(x0, t0, Dφ(x0, t0), D2φ(x0, t0)).

We can assume without loss of generality that the maximum is strict and global, andthat supt∈[0,T ] ‖φ(·, t)‖ < +∞. Let us set for simplicity xh = (x0)h and introducethe functions

fh : t 7→ h

[t/h]−1∑

l=0

Hh[1uh≥0](xh, lh, φ(·, t0))−∫ t

0

H [χ]∗(x0, s,Dφ(x0, t0), D2φ(x0, t0)) ds,

Two cases arise: if Dφ(x0, t0) 6= 0, then for almost every s ∈ [0, T ],

H [χ]∗(x0, s,Dφ(x0, t0), D2φ(x0, t0)) = H [χ](x0, s,Dφ(x0, t0), D

2φ(x0, t0)),

and by the consistency assumption (H3), we know that fhn(t) → 0 as n → +∞,locally uniformly for t ∈ [0, T ].

If Dφ(x0, t0) = 0, then a result by Barles and Georgelin [19] shows that we canalso assume that D2φ(x0, t0) = 0. In this case, H [χ]∗(x0, s,Dφ(x0, t0), D

2φ(x0, t0)) =

167


0 for almost every s ∈ [0, T ], thanks to (1.2). Assumption (H4) combined withthe fact that Hh[1uh≥0](xh, lh, 0) = 0 (assumption (H0) (i)) imply that for somemoduli of continuity mh,∣∣∣∣∣∣h

[t/h]−1∑

l=0

Hh[1uh≥0](xh, lh, φ(·, t0))

∣∣∣∣∣∣≤ T mh(|Dφ(xh, t0)| + |D2φ(xh, t0)|) −→

h→00,

because xh → x0, Dφ(xh, t0) → Dφ(x0, t0) = 0 and D2φ(xh, t0) → D2φ(x0, t0) = 0.In particular, fhn(t) → 0 as n→ +∞, locally uniformly for t ∈ [0, T ].

In both cases, the functions

vhn : (x, t) 7→ uhn(x, t) − φ(x, t) −∫ t

0

b(s) ds− fhn(t)

satisfy

lim sup∗(vhn)(x, t) = u(x, t) − φ(x, t) −∫ t

0

b(s) ds.

By a standard stability argument, there exists a subsequence of (hn), still denoted(hn) for simplicity, and a sequence (xn, knhn) → (x0, t0) of global maximum pointsof vhn with xn ∈ Πhn. We set

ξn = vhn(xn, knhn),

so that

uhn(x, t) ≤ φ(x, t) +

∫ t

0

b(s) ds+ fhn(t) + ξn (2.3)

for every (x, t) ∈ Πhn × 0, . . . , [T/hn]hn, with equality at (xn, knhn). Now thedefinition of the scheme (2.1) shows that if kn ≥ 1,

uhn(xn, knhn) = uhn(xn, (kn−1)hn)+hnHhn[1uhn≥0](xn, (kn−1)hn, uhn(·, (kn−1)hn)).

Replacing uhn in this expression thanks to (2.3), and using the assumption (H1) ofconditional monotonicity of the scheme, we therefore have

φ(xn, knhn) +

∫ knhn

0

b(s) ds+ fhn(knhn) + ξn

≤φ(xn, (kn − 1)hn) +

∫ (kn−1)hn

0

b(s) ds+ fhn((kn − 1)hn) + ξn

+ hnHhn[1uhn≥0](xn, (kn − 1)hn, φ(·, (kn − 1)hn) +

∫ (kn−1)hn

0

b(s)ds

+ fhn((kn − 1)hn) + ξn),

which, using assumption (H0) (i), reduces to

φ(xn, knhn) +

∫ knhn

0

b(s) ds+ fhn(knhn)

≤φ(xn, (kn − 1)hn) +

∫ (kn−1)hn

0

b(s) ds+ fhn((kn − 1)hn)

+hnHhn[1uhn≥0](xn, (kn − 1)hn, φ(·, (kn − 1)hn)).

168


Replacing fhn by its value, this transforms into

φ(xn, knhn) − φ(xn, (kn − 1)hn)

hn

≤ 1

hn

∫ knhn

(kn−1)hn

H [χ]∗(x0, s,Dφ(x0, t0), D

2φ(x0, t0)) − b(s)ds

+Hhn[1uhn≥0](xn, (kn − 1)hn, φ(·, (kn − 1)hn))

−Hhn[1uhn≥0](xhn, (kn − 1)hn, φ(·, t0)).

We now use the definition of G to deduce that


hn≤ 1

hn

∫ knhn

(kn−1)hn

G(x0, s,Dφ(x0, t0), D2φ(x0, t0)) ds

+Hhn[1uhn≥0](xn, (kn − 1)hn, φ(·, (kn − 1)hn))

−Hhn[1uhn≥0](xhn , (kn − 1)hn, φ(·, t0)).

Since φ and G are sufficiently regular, we have


hn− 1

hn

∫ knhn

(kn−1)hn

G(x0, s,Dφ(x0, t0), D2φ(x0, t0)) ds

−→n→+∞

φt(x0, t0) −G(x0, t0, Dφ(x0, t0), D2φ(x0, t0)).

To conclude, it therefore suffices to prove that

Hhn [1uhn≥0](xn, (kn−1)hn, φ(·, (kn−1)hn))−Hhn [1uhn≥0](xhn , (kn−1)hn, φ(·, t0))

has a nonpositive upper limit as n → +∞. But as n goes to +∞, xn → x0,xhn → x0, and φ(·, (kn − 1)hn) → φ(·, t0), so that thanks to assumption (H4), wehave for some moduli of continuity mhn ,

|Hhn[1uhn≥0](xn, (kn − 1)hn, φ(·, (kn − 1)hn))

−Hhn[1uhn≥0](xhn , (kn − 1)hn, φ(·, t0))|≤ mhn(|xn − xhn | + |Dφ(xn, (kn − 1)hn) −Dφ(xhn, t0)|

+ |D2φ(xn, (kn − 1)hn) −D2φ(xhn , t0)|),

which converges to 0 as n→ +∞, and the result follows.

Step 3. We just proved that u is a bounded L1-viscosity subsolution of (1.4), while uis a bounded L1-viscosity supersolution of (1.4). The comparison principle (A2) forthis equation then implies that u ≤ u in RN × [0, T ), while the converse inequalityis a direct consequence of their definition. This shows that in RN × [0, T ), u = ucoincide with the unique continuous L1-viscosity solution u of (1.4), and that (uhn)converges locally uniformly in RN × [0, T ) to u. Since of course we can extend H [χ]by 0 after time T , and use the previous argument on the extended time interval, wededuce that the convergence is in fact locally uniform in RN × [0, T ]. This finallyproves the convergence of (uhn, 1uhn≥0) to a couple (u, χ) which satisfies (1.4).

169


Moreover, χ being taken as the weak-∗ limit of (1uhn≥0), we can prove as in[17] that for almost all t ∈ [0, T ],

1u(·,t)>0 ≤ χ(·, t) ≤ 1u(·,t)≥0,

which means that (u, χ) also satisfies (1.5). In particular u is a weak solution of(1.1).

In fact, this proof shows that any sequence (uhn) of solutions of the scheme(2.1) admits a subsequence which converges locally uniformly to a weak solution of(1.1). As a consequence if this equation has a unique weak solution, then the wholesequence (uh) converges locally uniformly to the weak solution u of (1.1).

3 Applications

3.1 Dislocation dynamics

We are interested in particular in the dislocation dynamics equation (see thegeneral introduction, Chapters 2, 4 and the references therein), namely

ut = [c0(·, t) ⋆ 1u(·,t)≥0(x) + c1(x, t)]|Du| in RN × (0, T ),

u(·, 0) = u0 in RN ,(3.1)

where

c0(·, t) ⋆ 1u(·,t)≥0(x) =

∫

RN

c0(x− y, t) 1u(·,t)≥0(y) dy.

We assume that c0 and c1 satisfy the following assumptions, under which (A1) and(A2) are satisfied (see [16, 17]):

(D) (i) c0 ∈ C0([0, T ];L1(RN)), c1 ∈ C0(RN × [0, T ]; R).

(ii) For any t ∈ [0, T ], c0(·, t) is locally Lipschitz continuous and there exists aconstant C > 0 such that ‖Dc0‖L∞([0,T ];L1(RN )) ≤ C.

(iii) There exists a constant C > 0 such that, for any x, y ∈ RN and t ∈ [0, T ],

|c1(x, t)| ≤ C and |c1(x, t) − c1(y, t)| ≤ C|x− y|.Under these assumptions, there exists a weak solution of (3.1), as proved by Barles,Cardaliaguet, Ley and Monneau [16, Theorem 1.2] or with Barles, Cardaliaguet andLey in Chapter 2 or [17] (Theorem 3.2). We are going to study the convergence ofthe following approximation algorithm proposed by Alvarez, Carlini, Monneau andRouy [3] for N = 2, which is a particular case of (2.1). In [3], the authors prove shorttime existence of a classical viscosity solution to (3.1) and provide a convergence ratefor their scheme. We do not obtain such a rate but prove convergence of this schemeto a weak solution of (3.1) for long times. We set if x = xi1,...,iN ∈ Πh,

Hh[χ](x, kh, φ)

=

∑

j1,...,jN∈Z

c0(i1 − j1, . . . , iN − jN , k)χ(j1∆1, . . . , jN∆N , kh)

|Dh|(φ)(x)

+ c1(x, kh) |Dh|(φ)(x),

170

3. Applications

where

c0(m1, . . . , mN , k) =

∫

Qm1,...,mN

c0(y, kh) dy,

and |Dh|(φ)(x) is a monotone approximation of |Dφ(x)| adapted to the sign of thevelocity, such as the one proposed by Osher and Sethian [64] and used in [3]: let(e1, . . . , eN) denote the canonical basis of RN ; then for x ∈ Πh,

|Dh|(φ)(x) =

N∑

i=1

max

(φ(x+ ei) − φ(x)

∆i, 0

)2

+ min

(φ(x) − φ(x− ei)

∆i, 0

)21/2

if the sum of the nonlocal term and c1(x, kh) is nonnegative, and

|Dh|(φ)(x) =

N∑

i=1

min

(φ(x+ ei) − φ(x)

∆i, 0

)2

+ max

(φ(x) − φ(x− ei)

∆i, 0

)21/2

otherwise. In particular Hh satisfies (H0) with r = 1. Let M > 0 be such that

‖c0(·, t)‖L1(RN ) + |c1(x, t)| ≤M for any (x, t) ∈ RN × [0, T ].

The CFL condition to ensure the conditional monotonicity (H1) of the scheme is

√2N M

h

∆i

≤ 1 for any i = 1, . . . , N. (3.2)

The discrete convolution in the definition of Hh is efficiently computed using FastFourier Transform, see [3]. We now state our convergence result:

Theorem 3.1. Let c0 and c1 satisfy (D), and let u0 be a bounded and Lipschitzcontinuous function which satisfies (1.3). Let us fix space steps ∆i = λi h for anyi = 1, . . . , N , for some constants λi > 0 such that (3.2) holds.

Then there exists hn → 0 such that (uhn) converges locally uniformly to a weaksolution of (3.1) in RN × [0, T ].

If in addition we have

(D’) There exist c, c > 0 such that, for any x ∈ RN and t ∈ [0, T ],

|c0(x, t)| ≤ c,

0 < c ≤ −‖c0(·, t)‖L1(RN ) + c1(x, t) ≤ ‖c0(·, t)‖L1(RN ) + c1(x, t) ≤ c,

then the whole sequence (uh) converges locally uniformly in RN × [0, T ] to the unique

weak solution of (3.1).

Proof. We check the assumptions of Theorem 2.2, but will assume to avoid repetitionthat c1 = 0; the treatment of the term c1 is similar to – but easier than – thetreatment of the convolution term involving c0. To check assumptions (H2) to(H4), we first notice as in [3] that for x ∈ Πh and χ ∈ Bh(R

N × [0, T ]; [0, 1]),

Hh[χ](x, kh, φ) = c0(·, kh) ⋆ χ(·, kh)(x) |Dh|(φ)(x).

171


Assumption (H2) is satisfied with L = ‖u0‖∞, by a simple comparison with theconstant solutions ±‖u0‖∞. It only remains to prove assumptions (H3) and (H4).Let us pick x ∈ RN , φ ∈ C2

b (RN ; R), χh ∈ Bh(RN × [0, T ]; [0, 1]) such that χh χ

weakly-∗ in L∞(RN × [0, T ]; [0, 1]), and let us prove that

h

[t/h]−1∑

l=0

c0(·, lh) ⋆ χh(·, lh)(xh) |Dh|(φ)(xh) ds −→h→0

∫ t

0

c0(·, s) ⋆ χ(·, s)(x) |Dφ(x)| ds

locally uniformly for t ∈ [0, T ]. We decompose the difference of the two above termsas ∫ [t/h]h

t

c0(·, [s/h]h) ⋆ χh(·, s)(xh) |Dh|(φ)(xh) ds

+

∫ t

0

c0(·, [s/h]h) ⋆ χh(·, s)(xh) (|Dh|(φ)(xh) − |Dφ(x)|) ds

+|Dφ(x)|∫ t

0

c0(·, [s/h]h) ⋆ χh(·, s)(xh) − c0(·, s) ⋆ χh(·, s)(xh) ds

+|Dφ(x)|∫ t

0

c0(·, s) ⋆ χh(·, s)(xh) − c0(·, s) ⋆ χh(·, s)(x) ds

+|Dφ(x)|∫ t

0

c0(·, s) ⋆ χh(·, s)(x) − c0(·, s) ⋆ χ(·, s)(x) ds.

By definition of |Dh| and regularity of φ, the first term of this expression satisfies∣∣∣∣∣

∫ [t/h]h

t

c0(·, [s/h]h) ⋆ χh(·, s)(xh) |Dh|(φ)(xh) ds

∣∣∣∣∣

≤ |t− [t/h]h|M√

2N ‖Dφ‖∞ ≤M√

2N ‖Dφ‖∞ h,

while the second is estimated by∣∣∣∣∫ t

0

c0(·, [s/h]h) ⋆ χh(·, s)(xh) (|Dh|(φ)(xh) − |Dφ(x)|) ds∣∣∣∣

≤ T M | |Dh|(φ)(xh) − |Dφ(x)| | −→h→0

0.

The third term is, in absolute value, less than

|Dφ(x)|∫ t

0

‖c0(·, [s/h]h) − c0(·, s)‖L1(RN ) ds ≤ |Dφ(x)| T m(h),

where m is a modulus of continuity for c0 ∈ C0([0, T ];L1(RN)). We estimate thefourth term by

|Dφ(x)| T C |xh − x| ≤√N

2T C |Dφ(x)| (maxλi) h

using the facts that ‖Dc0‖L∞([0,T ];L1(RN )) ≤ C and

|xh − x|2 ≤N∑

i=1

(∆i

2

)2

=1

4

N∑

i=1

λ2i h

2 ≤ N

4(max λi)

2 h2.

172

3. Applications

Finally, the last term is equal to

|Dφ(x)|∫ t

0

∫

RN

c0(x− y, s) χh(y, s)− χ(y, s) dyds

which converges to 0 as h → 0 by definition of the weak-∗ convergence of (χh) toχ. This convergence is a priori merely pointwise in time but we notice as in [16,Remark 5.2] that the bound

∣∣∣∣∫

RN

c0(x− y, s)χh(y, s) dy

∣∣∣∣ ≤M

valid for any (x, s) ∈ RN × [0, T ] and h > 0 implies that the convergence is in fact

uniform, by Ascoli’s theorem.

To check (H4), let K be a compact set of RN and R be a positive constant, and

let us fix x, y ∈ K ∩ Πh, k ∈ N with kh ≤ T , φ, ψ ∈ C2b (R

N ; R) with ‖φ− ψ‖ ≤ R(‖ · ‖ is defined by (2.2)) and χ ∈ Bh(R

N × [0, T ]; [0, 1]). We want to prove that

|Hh[χ](x, kh, φ)−Hh[χ](y, kh, ψ)| ≤ mh(|x−y|, |Dφ(x)−Dψ(y)|+|D2φ(x)−D2ψ(y)|),

for some uniformly bounded moduli of continuity mh. To do this we write

Hh[χ](x, kh, φ) −Hh[χ](y, kh, ψ)

= c0(·, kh) ⋆ χ(·, kh)(x) |Dh|(φ)(x) − c0(·, kh) ⋆ χ(·, kh)(y) |Dh|(ψ)(y)

= c0(·, kh) ⋆ χ(·, kh)(x) |Dh|(φ)(x) − c0(·, kh) ⋆ χ(·, kh)(x) |Dφ(x)|+ c0(·, kh) ⋆ χ(·, kh)(x) |Dφ(x)| − c0(·, kh) ⋆ χ(·, kh)(x) |Dφ(y)|+ c0(·, kh) ⋆ χ(·, kh)(x) |Dφ(y)| − c0(·, kh) ⋆ χ(·, kh)(y) |Dφ(y)|+ c0(·, kh) ⋆ χ(·, kh)(y) |Dφ(y)| − c0(·, kh) ⋆ χ(·, kh)(y) |Dψ(y)|+ c0(·, kh) ⋆ χ(·, kh)(y) |Dψ(y)| − c0(·, kh) ⋆ χ(·, kh)(y) |Dh|(ψ)(y).

By definition of |Dh|, the first and the last terms of this equality are respectivelyestimated by

M | |Dh|(φ)(x) − |Dφ(x)| | ≤M

√2N

2‖D2φ‖∞ (max λi) h

and M | |Dh|(ψ)(y) − |Dψ(y)| | ≤M

√2N

2‖D2ψ‖∞ (maxλi) h

≤M

√2N

2(‖D2φ‖∞ +R) (maxλi) h.

The second term is easily dominated by

M N ‖D2φ‖∞ |x− y|

by regularity of φ, while the third term is, in absolute value, less than

C |x− y| ‖Dφ‖∞,

173


because ‖Dc0‖L∞([0,T ];L1(RN )) ≤ C. Finally, the fourth term is estimated by

M (|Dφ(y)| − |Dψ(y)|) ≤M |Dφ(x) −Dψ(y)| +M |Dφ(x) −Dφ(y)|≤M |Dφ(x) −Dψ(y)| +M N ‖D2φ‖∞ |x− y|.

This proves (H4) and concludes the proof of the first part of Theorem 3.1.

For the convergence of the entire sequence, we use the result of Chapter 4 or [18]which states that under assumptions (D) and (D’), then (3.1) has a unique weaksolution. The convergence of the whole sequence (uh) to this solution then followsfrom Theorem 2.2.

3.2 A FitzHugh-Nagumo type system

We are also interested in the following system presented in the general introduc-tion and studied in Chapters 2 and 4:

ut = α(v)|Du| in RN × (0, T ),

vt − ∆v = g+(v)1u≥0 + g−(v)(1 − 1u≥0) in RN × (0, T ),

u(·, 0) = u0, v(·, 0) = v0 in RN .

(3.3)

This system is obtained as the asymptotics as ε → 0 of the following FitzHugh-Nagumo system arising in neural wave propagation or chemical kinetics:

uε



(3.4)

in RN × (0, T ), where for (u, v) ∈ R2,f(u, v) = u(1 − u)(u− a) − v (0 < a < 1),

g(u, v) = u− γv (γ > 0).

The functions α, g+ and g− : R → R appearing in (3.3) are associated with f and g.This system has been studied in particular by Giga, Goto and Ishii [48] and Soravia,Souganidis [72]. They proved existence of a weak solution to (3.3). Moreover in [72],the convergence of the solution of (3.4) to a solution of (3.3) as ε→ 0 is proved.

If for χ ∈ L∞(RN × [0, T ]; [0, 1]), v denotes the solution ofvt − ∆v = g+(v)χ+ g−(v)(1 − χ) in RN × (0, T ),

v(·, 0) = v0 in RN ,(3.5)

and if c[χ](x, t) := α(v(x, t)), then Problem (3.3) reduces tout(x, t) = c[1u≥0](x, t)|Du(x, t)| in RN × (0, T ),

u(·, 0) = u0 in RN ,

(3.6)

174

3. Applications

which is a particular case of (1.1). In Chapter 2 or [17], with Barles, Cardaliaguetand Ley, we were therefore able to recover the existence result of [48, 72], and inChapter 4 or [18], we proved uniqueness in the case where α > δ in R for someδ > 0.

Let us state the assumptions satisfied by the data; they imply that (A1) and(A2) hold (see [17]):

(F) (i) α is Lipschitz continuous on R,(ii) g+ and g− are smooth on R

N , and there exist g and g in R such that

g ≤ g−(r) ≤ g+(r) ≤ g for all r in R.

We set γ = max|g|, |g|. Moreover we assume that

‖(g+)(i)‖∞ < +∞ and ‖(g−)(i)‖∞ < +∞ for i = 1, 2, 3.

(iii) v0 is of class C5 on RN with ‖Djv0‖∞ < +∞ for any j = 0, . . . , 5.Here we want to propose a numerical scheme to compute a weak solution, or the

weak solution if α > δ, of (3.3)-(3.6). To solve the heat equation part

vt − ∆v = g+(v)χ+ g−(v)(1 − χ),

we use an approximation scheme that we write in the following abstract form: webuild functions vh : RN × [0, T ] → R, such that vh is piecewise constant, i.e. forany (x, t) ∈ RN × [0, T ], vh(x, t) = vh(xh, [t/h]h), and such that for any k ∈ N with(k + 1)h ≤ T , for any x ∈ Πh,

vh(x, (k + 1)h) = Sh[χ](x, kh, vh),

vh(x, 0) = v0,h(x),(3.7)

where Sh[χ](x, kh, v) depends on χ(xi1,...,iN , lh)(i1,...,iN )∈ZN for l ∈ N up to k+1, andon vh(xi1,...,iN , lh)(i1,...,iN )∈ZN for l ∈ N up to k. Moreover v0,h is an approximationof the initial datum v0.

The scheme solving the heat equation being fixed, we then use our scheme (2.1)in the following form: for any k ∈ N such that (k + 1)h ≤ T , and for any x ∈ Πh,we set

uh(x, (k + 1)h) = uh(x, kh) + hα(vh(x, kh))|Dh|(uh(·, kh)),vh(x, (k + 1)h) = Sh[1uh≥0](x, kh, vh),

(3.8)

with the initial condition uh(x, 0) = u0(x)

vh(x, 0) = v0,h(x).

We recall that |Dh|(φ)(x) is the monotone approximation of |Dφ(x)| used in the pre-vious section. We easily see that this scheme is of the form (2.1) where H [χ](x, kh, u)depends on χ through all the values χ(·, lh) for 0 ≤ l ≤ k. We now formulate as-sumptions on the functions Sh which will guarantee convergence of (3.8) accordingto Theorem 2.2:

175


(S) (i) There exists M > 0 such that for any fixed χ ∈ Bh(RN × [0, T ]; [0, 1]),

the solution vh of (3.7) satisfies, for any x ∈ Πh and k ∈ N with kh ≤ T ,

|vh(x, kh)| ≤M independently of h.

(ii) If χh ∈ Bh(RN ×[0, T ]; [0, 1]) is such that χh χ in L∞(RN ×[0, T ]; [0, 1])

for the weak-∗ topology as h → 0, then the solution vh of (3.7) associated to χh

converges pointwise to the solution v of (3.5) in B(0, R) × [0, T ], where we setR = R0 + T

√N max λi and R0 is given by (1.3).

(iii) For any compact subset K of RN , there exist uniformly bounded moduli

of continuity mh such that for any h > 0, x, y ∈ K ∩ Πh, any k, h > 0 with kh ≤ Tand χ ∈ Bh(R

N × [0, T ]; [0, 1]), the solution vh of (3.7) satisfies

|vh(x, kh) − vh(y, kh)| ≤ mh(|x− y|),

and such that mh(η) → 0 as h, η → 0.

Our convergence result is the following:

Theorem 3.2. Assume that α, g+, g− and v0 satisfy (F), while u0 is a bounded andLipschitz continuous function which satisfies (1.3). Let uh be defined by the scheme(3.8) such that (S) holds and the ∆i’s satisfy

√2N max|α(r)|, |r| ≤M h

∆i≤ 1 for any i = 1, . . . , N, (3.9)

where M is the constant given by assumption (S) (i). Then there exists hn → 0 suchthat (uhn) converges locally uniformly in RN × [0, T ] to a weak solution of (3.6).

If in addition there exists δ > 0 such that α(r) ≥ δ for any r ∈ R, then thewhole sequence (uh) converges locally uniformly in RN × [0, T ] to the weak solutionof (3.6).

Proof. Assumption (S) (i) guarantees the existence of a constant M such that forany fixed χ ∈ Bh(R

N×[0, T ]; [0, 1]), the solution vh of (3.7) satisfies, for any x, y ∈ Πh

and k ∈ N with kh ≤ T ,

|vh(x, kh)| ≤M independently of h.

The CFL condition to ensure the conditional monotonicity of the first part of thescheme (3.8) is exactly (3.9), while the stability of this scheme follows as in thedislocation case. It only remains to check assumptions (H3) and (H4) of Theorem2.2. This verification is very similar to the above proof in the dislocation case: ituses assumption (S) and the Lipschitz continuity of α. As a consequence, Theorem2.2 guarantees the existence of a subsequence (uhn) converging locally uniformly inRN × [0, T ] to a weak solution of (3.6).

If in addition there exists δ > 0 such that α(r) ≥ δ for any r ∈ R, then (3.6)has a unique weak solution (see Chapter 4 or [18]). The convergence of the wholesequence (uh) to this solution follows once more from Theorem 2.2.

176

3. Applications

Let us now give an example of scheme (3.7) which satisfies (S). Due to the lackof regularity of the function χ, we will solve an approximate equation in which theterm χ is regularized by convolution: for ε ∈ (0, 1), let (ρε) be a mollifier on RN ×R

such that Supp(ρε) ⊂ [−ε, ε]N+1, ρε(−x,−t) = ρε(x, t) ≥ 0 for all (x, t) ∈ RN × R,‖ρε‖1 = 1 and∥∥∥∥∂i

∂tiDjρε

∥∥∥∥1

≤ A

ε(i+j)(N+1)for (i, j) = (2, 0) or (i = 0, 1 and i+ j ≤ 3), (3.10)

for some constant A > 0. To ensure that our scheme is nonanticipative, we shift ρε

in time by ε and set

χε(x, t) =

∫ T

0

∫

RN

ρε(x− y, t− s− ε)χ(y, s) dyds.

We are going to solve (3.5) by the standard forward Euler scheme, with the regular-ization χε of χ. This regularization is essential to obtain estimates on the solutionvε

h, and pass to the limit thanks to a good choice of balance between ε and h.

Let us fix the space steps ∆i by the relation ∆i = λi h for some fixed constantsλi > 0 to be made precise later. Recall that these conditions are essential to guaran-tee compactness of the front uh(·, t) ≥ 0 for any time t ∈ [0, T ], when uh satisfies(2.1). However, for the forward Euler scheme to be stable and monotone, theseconditions are not adapted.

For this reason, we need to solve (3.5) on a refined time grid: let h′ be anothertime step such that h/h′ =: p ∈ N∗; the integer p may depend on h. We define theoperator T kh′

h′ [χ] corresponding to the k-th step of the forward Euler scheme for (3.5)on this refined grid; that is, for any function v : Πh → R, χ ∈ Bh(R

N × [0, T ]; [0, 1]),for any x ∈ Πh and k, h′ such that (k + 1)h′ ≤ T ,

T kh′

h′ [χ](v)(x) = v(x) + h′N∑

i=1

v(x+ ∆iei) − 2v(x) + v(x− ∆iei)

∆2i

+ h′ g+(v(x))χε(x, kh′) + h g−(v(x))(1 − χε(x, kh′)), (3.11)

where (e1, . . . , eN) is the canonical basis of RN .

Then we set for any v : Πh → R, χ ∈ Bh(RN × [0, T ]; [0, 1]), for any x ∈ Πh and

k, h such that (k + 1)h ≤ T ,

Sh[χ](x, kh, v) = Tkh+(p−1)h′

h′ [χ] · · · T kh+h′

h′ [χ] T khh′ [χ](v)(x), (3.12)

and we denote by vεh the solution of (3.7) with initial condition

v0,h(x) = vε0(x) (3.13)

for some regularization vε0 of v0 of class C∞ with ‖Djvε

0‖∞ ≤ ‖Djv0‖∞ for anyj = 0, . . . , 5, and such that vε

0 → v0 uniformly as ε→ 0.

177


This means that, to define vεh(x, (k + 1)h) knowing vε

h(x, kh), we split the timeinterval [kh, (k+ 1)h] in p = p(h) intervals of length h′ and make p iterations of theoperator Th′ , starting from vε

h(x, kh).

To explain the choice of h′, we notice that the linear part of (3.11), which isrepresented by the operator

G(h′) : v = (v(x))x∈Πh7→(v(x) + h′

N∑

i=1

v(x+ ∆iei) − 2v(x) + v(x− ∆iei)

∆2i

)

x∈Πh

,

is monotone and satisfies‖G(h′)v‖∞ ≤ ‖v‖∞

under the CFL condition

maxh′

∆2i

≤ 1

2N. (3.14)

Since in addition we have for any k, h′ such that kh′ ≤ T ,

|h′ g+(vεh(x, kh

′))χε(x, kh′) + h′ g−(vεh(x, kh

′))(1 − χε(x, kh′))| ≤ γ h′,

it is easy to see that under condition (3.14), for any h and ε we have

‖vεh‖∞ ≤ ‖v0‖∞ + γ T = M.

We therefore choose our time step h′ by the relation ∆i = µi

√h′ for some constant

µi > 0 such that h/h′ ∈ N∗ and (3.14) holds: more precisely, we fix constants

µi ≥√

2N independent of h such that λi/µi =: ν does not depend on i, and set

h′ = (νh)2, where h =1

ν2p(3.15)

for some p ∈ N∗. For this particular scheme, we have the following convergenceresult:

Proposition 3.3. Assume that α, g+, g− and v0 satisfy (F), while u0 is a boundedand Lipschitz continuous function which satisfies (1.3). Let us fix ∆i = λi h forsome fixed constants λi > 0 such that (3.9) holds with

M = ‖v0‖∞ + γ T,

and let us define h′ by (3.15). We also assume that ε is linked to h by the relation

ε3(N+1) = h2β (3.16)

for some fixed β ∈ (0, 1).

Let us define the scheme (3.7) with Sh and v0,h defined by (3.11), (3.12) and(3.13). Then the assumptions of Theorem 3.2 are satisfied.

178

3. Applications

Proof. First of all, as explained above, (S) (i) is satisfied with M = ‖v0‖∞ + γ T ,and the ∆i’s were chosen so as to satisfy (3.9) with this M .

To check (S) (ii), let us fix a sequence of functions χh ∈ Bh(RN × [0, T ]; [0, 1])

such that χh χ weakly-∗ in L∞(RN×[0, T ]; [0, 1]) as h→ 0. We want to prove thatfor the choice of ε(h) given by (3.16), the solution vε

h of (3.7) associated to χh withinitial condition vε

0 converges pointwise to the solution v of (3.5) in B(0, R)× [0, T ]as h→ 0. To do so, we set χε

h := (χh)ε and write

vεh − v = (vε

h − wεh) + (wε

h − wh) + (wh − v),

where wh (resp. wεh) denotes the solution of (3.5) associated to χh (resp. χε

h) withinitial condition v0 (resp. vε

0). That is, we split the error into three parts, the firstpart concerning the approximation error coming from the scheme, but with regularsource terms χε

h, the second part taking into account the error on exact solutions of(3.5), but as we relax the regularity of χε by letting χε

h → χh, and the third partdealing with the weak convergence of χh to χ.

Step 1: the term vεh − wε

h. Let us set

Ek = (Ek(x))x∈Πh:= (vε

h(x, kh′) − wε

h(x, kh′))x∈Πh

to be the approximation error at step k. Let us also set ek = (ek(x))x∈Πh, where

ek(x) :=wε

h(x, (k + 1)h′) −G(h′)wεh(x, kh

′)

h′

− g+(wεh(x, kh

′))χεh(x, kh

′) − g−(wεh(x, kh

′))(1 − χεh(x, kh

′)),

which represents the consistency error of the scheme. Classical error estimates onthe explicit Euler scheme for the heat equation imply that there exists a constantC > 0 such that for any x ∈ Πh and k, h′ with kh′ ≤ T ,

|ek(x)| ≤ C

ε3(N+1)(h′ + max ∆2

i ). (3.17)

Indeed, the Hölder theory for parabolic equations (see for example [52]) shows that∥∥∥∥∂2wε

h

∂t2

∥∥∥∥∞

≤ A

ε3(N+1)and

∥∥D4wεh

∥∥∞ ≤ A

ε3(N+1)

for some constant A > 0, thanks to (3.10) and the bounds on the derivatives of g+,g− and the initial datum v0. Then we remark that

Ek+1(x)

= vεh(x, (k + 1)h′) − wε

h(x, (k + 1)h′)

=G(h′)vεh(·, kh′)(x) + h′ g+(vε

h(x, kh′))χε

h(x, kh′) + h′ g−(vε

h(x, kh′))(1 − χε

h(x, kh′))

−G(h′)wεh(·, kh′)(x) − h′ g+(wε

h(x, kh′))χε

h(x, kh′) − h′ g−(wε

h(x, kh′))(1 − χε

h(x, kh′))

− h′ ek(x),

179


which we rewrite as

Ek+1(x) = G(h′)[vεh(·, kh′) − wε

h(·, kh′)](x) − h′ ek(x)

+ h′ [g+(vεh(x, kh

′)) − g+(wεh(x, kh

′))]χεh(x, kh

′)

+ h′ [g−(vεh(x, kh

′)) − g−(wεh(x, kh

′))](1 − χεh(x, kh

′)).

If D denotes a Lipschitz constant for g+ and g−, then we obtain, using the fact that‖G(h′)‖ ≤ 1,

‖Ek+1‖∞ ≤ ‖Ek‖∞ + h′‖ek‖∞ +Dh′ ‖Ek‖∞ = (1 +Dh′)‖Ek‖∞ + h′ ‖ek‖∞.By induction, and using the fact that E0 = 0, we easily deduce that for any k withkh′ ≤ T ,

‖Ek‖∞ ≤ h′k∑

i=0

(1 +Dh′)i ‖ek−i‖∞.

Using (3.17), we obtain that for any k with kh′ ≤ T ,

‖Ek‖∞ ≤ TeDT C

ε3(N+1)(h′ + max ∆2

i )

≤ TeDT C

ε3(N+1)(1 + maxµ2

i ) ν2 h2, (3.18)

thanks to the choices of ∆i = µi

√h′ and h′ = (νh)2. We therefore see that if we

choose ε as in (3.16), i.e. ε3(N+1) = h2β for some β ∈ (0, 1), then vεh − wε

h convergesto 0 uniformly on Πh as h → 0. Moreover, an easy consequence of the explicitresolution of (3.5) (see Lemma 4.5 in Chapter 2) is that there exists a constantkN > 0 depending only on N such that for any x, y ∈ R

N ,

|wεh(x, kh) − wε

h(y, kh)| ≤(‖Dv0‖∞ + kN γ

√T)|x− y|.

As a consequence, vεh − wε

h also converges to 0 uniformly on RN as h→ 0.

Step 2: the term wεh−wh. Let us first prove that χε

h−χh 0 in L∞(RN×[0, T ]; [0, 1])weakly-∗ as h→ 0. For any φ ∈ L1(RN × [0, T ]; R),

∫ T

0

∫

RN

χεh(x, t)φ(x, t) dxdt−

∫ T

0

∫

RN

χh(x, t)φ(x, t) dxdt

=

∫ T

0

∫

RN

(∫ T

0

∫

RN

χh(y, s) ρε(x− y, t− s− ε) dyds

)φ(x, t) dxdt

−∫ T

0

∫

RN

χh(x, t)φ(x, t) dxdt.

Exchanging the variables (x, t) and (y, s) in the first integral, which is permitted bythe facts that χh takes values in [0, 1], and that ρε and φ ∈ L1, we transform thisdifference of integrals into

∫ T

0

∫

RN

χh(y, s)

(∫ T

0

∫

RN

ρε(x− y, t− s− ε)φ(x, t) dxdt

)dyds

−∫ T

0

∫

RN

χh(y, s)φ(y, s) dyds

180

3. Applications

which, in absolute value, is less than∫ T

0

∫

RN

∣∣∣∣(∫ T

0

∫

RN

ρε(x− y, t− s− ε)φ(x, t) dxdt

)− φ(y, s)

∣∣∣∣ dyds,

since |χh| ≤ 1. Using the fact that ρε is symmetric, this integral is equal to∫ T

0

∫

RN

∣∣∣∣(∫ T

0

∫

RN

ρε(y − x, s− t+ ε)φ(x, t) dxdt

)− φ(y, s)

∣∣∣∣ dyds,

that is to say,‖ρε(·, · + ε) ⋆ φ− φ‖L1(RN×[0,T ]),

where φ is the extension of φ to RN ×R by φ(·, t) = 0 if t /∈ [0, T ]. Reproducing thestandard proof on approximation by convolution (using the approximation of φ bya function of class C1), we see that this term converges to 0 as ε = ε(h) → 0. Thisproves the claim.

We deduce from this assertion and the fact that vε0 → v0 uniformly, that wε

h−wh

converges locally uniformly to 0 as h→ 0. This verification is similar to the proof ofTheorem 4.4 of Chapter 2, or [17], based on the explicit resolution of (3.5) in termsof the Green function of the heat equation.

Step 3: the term wh − v. We prove in the same manner that this term convergeslocally uniformly to 0 as h → 0, since χh χ weakly-∗ in L∞(RN × [0, T ]; [0, 1]).This concludes the verification of (S) (ii).

Let us finally check (S) (iii) for the choice of ε given by (3.16): let K be acompact subset of RN , let x, y ∈ K ∩ Πh, kh ≤ T and χ ∈ Bh(R

N × [0, T ]; [0, 1]).To estimate vε

h(x, kh) − vεh(y, kh), where vε

h is the solution of (3.7), we write

vεh(x, kh) − vε

h(y, kh) = (vεh(x, kh) − wε

h(x, kh)) + (wεh(x, kh) − wε

h(y, kh))

+ (wεh(y, kh) − vε

h(y, kh)).

Using the error estimate (3.18), we know that

|vεh(x, kh)−wε

h(x, kh)|+ |wεh(y, kh)− vε

h(y, kh)| ≤ 2T eDT C

ε3(N+1)(1+ maxµ2

i ) ν2 h2.

Moreover, as recalled above, the solution wεh of (3.5) associated to χε

h satisfies

|wεh(x, kh) − wε

h(y, kh)| ≤(‖Dv0‖∞ + kN γ

√T)|x− y|.

With the previous choice of ε, we therefore obtain that (S) (iii) is satisfied with

mh(η) = 2T eDT C (1 + maxµ2i ) ν

2 h2(1−β) +(‖Dv0‖∞ + kN γ

√T)η.

This concludes the proof of Proposition 3.3 and implies the convergence of ourscheme according to Theorem 3.2.

Aknowledgment. I would like to thank Pierre Cardaliaguet for his kind advice andsupport during the preparation of the article presented in this chapter. This workwas partially supported by the ANR (Agence Nationale de la Recherche) throughMICA project (ANR-06-BLAN-0082).

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187

Contribution à l’étude d’équations de propagations de fronts locales et non-locales

Résumé : Ce travail porte sur l’étude de propagations de fronts gouvernées par des lois locales et

non-locales. Dans la méthode par lignes de niveau, le front est vu comme ligne de niveau 0 d’une fonction

auxiliaire. A la loi géométrique d’évolution du front correspond alors une équation de Hamilton-Jacobi sur

cette fonction, que nous envisageons dans le cadre des solutions de viscosité.

Dans les modèles non-locaux, la difficulté principale pour prouver des résultats d’existence ou d’unicité

est l’absence de principe d’inclusion entre les fronts. Dans la mé thode par lignes de niveau, ceci correspond

à une absence de principe de comparaison entre les fonctions, qui rend impossible l’utilisation des techniques

habituelles. L’utilisation alternative de méthodes de point fixe associe à toute équation non-locale une

famille d’é quations locales. La compréhension de la régularité des solutions des équations locales, et en

particulier du pé rimètre de leurs lignes de niveau, apparaît alors cruciale dans les arguments de point fixe.

Dans le chapitre 1, on prouve des formulations intégrales de l’équation eikonale locale, dont on déduit

des estimations sur le périmètre des lignes de niveau de ses solutions.

Dans le reste des travaux, on s’intéresse aux équations non-locales, et notamment à une notion de

solution faible pour ces équations. Deux modèles non-locaux, la dynamique des dislocations et un système

de type FitzHugh-Nagumo, sont également étudiés en détails. Le chapitre 2 concerne l’existence de solutions

faibles pour des équations non-locales générales, tandis que dans le chapitre 3, le résultat est affiné pour

la dynamique des dislocations avec un terme de courbure moyenne, qui possède un effet régularisant et

permet l’utilisation de techniques variationnelles. Dans le chapitre 4, on prouve l’unicité de solution pour

les deux modèles lorsque la vitesse est positive. Enfin le chapitre 5 est consacré à l’approximation numé

rique des lois non-locales, et notamment à la preuve d’un résultat de convergence de schémas généraux

vers une solution faible. Ce résultat est appliqué à un exemple explicite de schéma pour chacune des deux

équations modèle.

Contribution to the study of local and nonlocal front propagation equations

Abstract : The subject of this work is the study of front propagations governed by local and nonlocal

laws. In the so-called level-set method, the front is seen as the 0 level-set of an auxiliary function. In this

context, the geometric evolution law of the front corresponds to a Hamilton-Jacobi equation satisfied by

this function ; this equation is considered in the framework of viscosity solutions.

In nonlocal models, the main obstacle to existence and uniqueness results is the absence of inclusion

principle between fronts. In the level-set method, this corresponds to an absence of comparison principle

between functions, which makes impossible the use of classical techniques. The alternative use of fixed point

methods associates to any nonlocal equation a family of local equations. Understanding the regularity of

the solutions of local equations, and in particular the perimeter of their level-sets, therefore appears crucial

in fixed point type arguments.

In Chapter 1, we prove integral formulations for the local eikonal equation, from which we deduce

perimeter estimates on the level-sets of its solutions.

In the rest of this work, we focus on nonlocal equations, and in particular on a notion of weak solution

for these equations. Two nonlocal models, the dislocation dynamics equation and a FitzHugh-Nagumo

type system, are also studied in details. Chapter 2 is concerned with the existence of weak solutions for

general nonlocal equations, while in Chapter 3, we refine the result in the case of the dislocation dynamics

equation with a mean curvature term, which has a regularizing effect and enables the use of variationnal

techniques. In Chapter 4, we prove uniqueness of a solution for the two models in the case of positive

velocities. Finally, Chapter 5 is devoted to the numerical approximation of nonlocal laws, and in particular

to the proof of a result of convergence of general schemes to a weak solution. This result is applied to an

explicit example of scheme for each of the two model equations.

thèse de doctorat de mathématiques présentée à par...

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