priority programme 1962 hyperbolic maxwell variational ... · edited by spp1962 at weierstrass...

Priority Programme 1962

Hyperbolic Maxwell Variational Inequalities of the Second Kind

Irwin Yousept

Non-smooth and Complementarity-basedDistributed Parameter Systems:Simulation and Hierarchical Optimization

Preprint Number SPP1962-069

received on August 22, 2018

Edited bySPP1962 at Weierstrass Institute for Applied Analysis and Stochastics (WIAS)

Leibniz Institute in the Forschungsverbund Berlin e.V.Mohrenstraße 39, 10117 Berlin, Germany

E-Mail: [email protected]

World Wide Web: http://spp1962.wias-berlin.de/

http://spp1962.wias-berlin.de/

HYPERBOLIC MAXWELL VARIATIONAL INEQUALITIESOF THE SECOND KIND

IRWIN YOUSEPT ∗

Abstract. In this note, we provide an existence and uniqueness theorem for the weak andstrong solutions to a general class of hyperbolic Maxwell variational inequalities of the second kind.The tools for our analysis are the semigroup theory for Maxwell’s equations, the Moreau-Yosidaregularization theory and the subdifferential analysis.

Key words. hyperbolic Maxwell variational inequalities of the second kind, well-posedness.

AMS subject classifications. 35Q60, 35L85.

1. Introduction. Physical phenomena in electromagnetism can lead to hyper-bolic variational inequalities with a Maxwell structure. They include applicationproblems arising from electromagnetic processes in polarizable media and Bean’scritical-state model in high-temperature superconductivity. The very first study onhyperbolic variational inequalities in electromagnetism goes back to Duvaut and Li-ons [9]. By employing the Galerkin-ansatz and a parabolic regularization technique,they [9, Section 8] proved an existence and uniqueness result for the strong solutionto a hyperbolic obstacle-type Maxwell variational inequality (VI of the first kind).Some years later, Milani [14,15] extended their results to the case of time-dependentobstacle set. Quite recently, the author [24] shown that the Bean critical-state modelfor high-temperature superconductors [8] leads to a hyperbolic Maxwell variational in-equality of the second kind. The optimal control problem of this variational inequalitywas analyzed in [25] (cf. also [23]). We note that the analysis of the Bean critical-state model governed by the eddy current equations goes back to Prigozhin [19–21]and Barrett and Prigozhin [4–7]. The corresponding model governed by the Maxwellequations was considered in [11,12]. We also refer the reader to [2,16,17] for parabolicand elliptic variational inequalities of a p-curl type arising from the power law for theeddy current equations.

This note analyzes a general class of hyperbolic Maxwell variational inequalitiesof the second kind on an open set Ω ⊆ R3 representing the physical medium, whereelectromagnetic fields are acting. Our main result is the existence and uniquenesstheorem for the strong and weak solutions. The main tools for our analysis are thesemigroup theory for Maxwell’s equations, the Moreau-Yosida approximation theoryand the set-valued subdifferential analysis. Our paper is organized as follows. In theupcoming section, we introduce all the function spaces used in our analysis, includingthe definition of the Maxwell operator. Some well-known properties of the Maxwelloperator are also mentioned in this section. In Section 3, we present our main existenceand uniqueness theorem. The proof is presented in Sections 4 and 5.

2. Preliminaries. We introduce the Hilbert space

H(curl) :=q ∈ L2(Ω)

∣∣ curl q ∈ L2(Ω),

∗Universität Duisburg-Essen, Fakultät für Mathematik, Thea-Leymann-Str. 9, D-45127 Essen,Germany, [email protected]. This work was supported by the German Research FoundationPriority Programm DFG SPP 1962 "Non-smooth and Complementarity-based Distributed ParameterSystems: Simulation and Hierarchical Optimization", Project YO 159/2-1.

1

2 HYPERBOLIC MAXWELL VARIATIONAL INEQUALITIES OF THE SECOND KIND

where the curl -operator is understood in the sense of distributions. Note that, for agiven Hilbert space V , we use the notation ‖·‖V and (·, ·)V for a standard norm and astandard scalar product in V. A bold typeface is used to indicate a three-dimensionalvector function or a Hilbert space of three-dimensional vector functions. As usual,C∞0 (Ω) stands for the space of all infinitely differentiable three-dimensional vectorfunctions with compact support contained in Ω. We denote the closure of C∞0 (Ω)with respect to the H(curl)-topology by

H0(curl) := C∞0 (Ω)‖·‖H(curl)

.

It is well known that the Hilbert spaceH0(curl) admits the following characterization:

(2.1) H0(curl) =q ∈H(curl) | (q, curlv)L2(Ω) = (curl q,v)L2(Ω) ∀v ∈H(curl)

.

Let ε, µ : Ω→ R3×3 denote the electric permittivity and the magnetic permeability inthe medium Ω. They are assumed to be of class L∞(Ω)3×3, symmetric and uniformlypositive-definite in the sense that there exist constants ε, µ > 0 such that

(2.2) ξT ε(x)ξ ≥ ε|ξ|2 and ξTµ(x)ξ ≥ µ|ξ|2 for a.e. x ∈ Ω and all ξ ∈ R3.

For symmetric and uniformly positive definite matrix-valued function α ∈ L∞(Ω)3×3,let L2

α(Ω) denote the weighted L2(Ω)-space endowed with the weighted scalar product(α·, ·)L2(Ω). Based on this notation, let us define the weighted product space

X := L2ε(Ω)×L2

µ(Ω),

equipped with the scalar product

(2.3) ((e,h), (v,w))X = (εe,v)L2(Ω) + (µh,w)L2(Ω), ∀(e,h), (v,w) ∈X.

Now, we introduce the (unbounded) Maxwell operator

(2.4) A : D(A) ⊂X →X, A := −(ε 00 µ

)−1(0 −curl

curl 0

),

withD(A) := H0(curl)×H(curl). Due to the choice of the weighted Hilbert spaceXand (2.1), the Maxwell operator A : D(A) ⊂ X → X is skew-adjoint, i.e., D(A∗) =D(A) and A∗ = −A. Therefore, by virtue of Stone’s theorem [18, Theorem 10.8, p.41], A generates a strongly continuous group Ttt∈R of unitary operators on X.

Lemma 2.1. Let Stt∈R be a strongly continuous group of unitary operatorson X. Furthermore, suppose that (e,h) ∈ C([0, T ],X), (e0,h0) ∈ X and (w, w) ∈L1((0, T ),X) satisfy the variation of constants formula

(e,h)(t) = St(e0,h0) +

∫ t

0

St−s(w, w)(s) ds, ∀t ∈ [0, T ].

Then, the following energy balance equality holds:

∥∥(e,h)(t)∥∥2

X=∥∥(e0,h0)

∥∥2

X+ 2

∫ t

0

((w, w)(s), (e,h)(s))X ds, ∀t ∈ [0, T ].

IRWIN YOUSEPT 3

Proof. As Stt∈R is a strongly continuous group of unitary operators on X,its infinitesimal generator B : D(B) ⊂ X → X is skew-adjoint (Stone’s theo-rem). Since D(B) ⊂ X and C∞0 ((0, T ),X) ⊂ L1((0, T ),X) are dense, there exist(en,0,hn,0)∞n=1 ⊂ D(B) and (wn, wn)∞n=1 ⊂ C∞0 ((0, T ),X) such that

(2.5) limn→∞

‖(en,0−e0,hn,0−h0)‖X = 0 and limn→∞

‖(wn−w, wn−w)‖L1((0,T ),X) = 0.

For every n ∈ N, we define

(en,hn)(t) := St(en,0,hn,0) +

∫ t

0

St−s(wn, wn)(s) ds, ∀t ∈ [0, T ].

By definition and since Stt∈R is unitary, we infer that

‖(en − e,hn − h)(t)‖X

=

∥∥∥∥St(en,0 − e0,hn,0 − h0) +

∫ t

0

St−s(wn −w, wn − w)(s) ds

∥∥∥∥X

≤ ‖(en,0 − e0,hn,0 − h0)‖X + ‖(wn −w, wn − w)‖L1((0,t),X), ∀t ∈ [0, T ], ∀n ∈ N.

It follows therefore for every n ∈ N that

‖(en − e,hn − h)‖C([0,T ],X) ≤ ‖(en,0 − e0,hn,0 − h0)‖X + ‖(wn −w, wn − w)‖L1((0,T ),X),

and so (2.5) implies

(2.6) limn→∞

‖(en − e,hn − h)‖C([0,T ],X) = 0.

On the other hand, since (en,0,hn,0) ∈ D(B) and (wn, wn) ∈ C∞0 ((0, T ),X), it holdsfor every n ∈ N that (en,hn) ∈ C([0, T ], D(B)) ∩ C1([0, T ],X), and it is exactly thesolution of

d

dt(en,hn)(t) = B(en,hn)(t) + (wn, wn)(t), ∀t ∈ [0, T ],

(en,hn)(0) = (en,0,hn,0).

See [18, Corollary 2.5, p. 107] for this classical result. Thus, for every t ∈ [0, T ] andn ∈ N, it follows that∫ t

0

(d

dt(en,hn)(s), (en,hn)(s)

)X

ds

=

∫ t

0

(B(en,hn)(s), (en,hn)(s))X ds+

∫ t

0

((wn, wn)(s), (en,hn)(s))X ds

=

∫ t

0

((wn, wn)(s), (en,hn)(s))X ds,

since B is skew-adjoint. In conclusion, we obtain for every t ∈ [0, T ] and n ∈ N that

1

2‖(en,hn)(t)‖2X −

1

2‖(en,0,hn,0)‖2X =

∫ t

0

((wn, wn)(s), (en,hn)(s))X ds.

Now, passing to the limit n→∞, the assertion follows from (2.5) and (2.6).


3. Main results. In this section, we present our existence and uniqueness resultsfor a general class of hyperbolic Maxwell variational inequalities of the second kind.The proof is given in Section 4 and Section 5. In the following, let T ∈ R+ and

ϕ : X → R := R ∪ +∞

be a proper, convex and lower semicontinuous (l.s.c.) function. We recall that thesubdifferential ∂ϕ : X → 2X is a set-valued operator, where, for every (e,h) ∈ X,∂ϕ(e,h) contains all subgradients of ϕ at (e,h), i.e.,

(3.1)∂ϕ(e,h) =

(e, h) ∈X | ((e, h), (v,w)− (e,h))X ≤ ϕ(v,w)− ϕ(e,h)

∀(v,w) ∈X.

Assumption 3.1. For every M > 0, there exists a constant CM > 0 such that

‖(e, h)‖X ≤ CM , ∀(e, h) ∈ ∂ϕ(e,h),

for all (e,h) ∈X satisfying ‖(e,h)‖X ≤M .Theorem 3.2. Let ϕ : X → R be a proper, convex and l.s.c. function sat-

isfying ∂ϕ(0, 0) 6= ∅ and Assumption 3.1. Then, for every (f ,g) ∈ L1((0, T ),X)

and (E0,H0) ∈ X, there exist unique pairs (E,H) ∈ C([0, T ],X) and (E, H) ∈L∞((0, T ),X) such that

(3.2)

d

dt

∫Ω

εE(t) · v + µH(t) ·w dx+

∫Ω

E(t) · curlw −H(t) · curlv dx

+

∫Ω

εE(t) · v + µH(t) ·w dx =

∫Ω

f(t) · v + g(t) ·w dx,

for a.e. t ∈ (0, T ) and all (v,w) ∈H0(curl)×H(curl),

(E, H)(t) ∈ ∂ϕ((E,H)(t)) for a.e. t ∈ (0, T ),

(E,H)(0) = (E0,H0),

and, for every (v,w) ∈H0(curl)×H(curl), the mapping t 7→ ((E,H)(t), (v,w))Xis absolutely continuous from [0, T ] to R. Moreover, (E,H) satisfies the Maxwellvariational inequality

(3.3)

d

dt

∫Ω

εE(t) · (v − 1

2E(t)) + µH(t) · (w − 1

2H(t)) dx

+

∫Ω

E(t) · curlw −H(t) · curlv dx+ ϕ(v,w)− ϕ((E,H)(t))

≥∫

Ω

f(t) · (v −E(t)) dx+

∫Ω

g(t) · (w −H(t)) dx,


(E,H)(0) = (E0,H0).

Theorem 3.3. Let ϕ : X → R be a proper, convex and l.s.c. function satisfy-ing ∂ϕ(0, 0) 6= ∅ and Assumption 3.1. Then, for every (f ,g) ∈ C0,1([0, T ],X) and

IRWIN YOUSEPT 5

(E0,H0) ∈H0(curl)×H(curl), the Maxwell variational inequality

(VI)

∫Ω

εd

dtE(t) · (v −E(t)) + µ

d

dtH(t) · (w −H(t)) dx

+

∫Ω

curlE(t) ·w − curlH(t) · v dx+ ϕ(v,w)− ϕ((E,H)(t))

≥∫

Ω

f(t) · (v −E(t)) dx+

∫Ω

g(t) · (w −H(t)) dx,

for a.e. t ∈ (0, T ) and all (v,w) ∈X,

(E,H)(0) = (E0,H0)

admits a unique solution (E,H) ∈ L∞((0, T ),H0(curl)×H(curl))∩W 1,∞((0, T ),X),which at the same time solves both (3.2) and (3.3).

Remark 3.4. If ϕ ≡ 0 and g ≡ 0, then Theorem 3.2 and Theorem 3.3 arenothing but the classical existence and uniqueness results (cf. Leis [13]) for the weakand strong solutions to the Maxwell equations

εd

dtE − curlH = f in Ω× (0, T ), E(0) = E0 in Ω,

µd

dtH + curlE = 0 in Ω× (0, T ), H(0) = H0 in Ω.

4. Proof of Theorem 3.2. We split our proof for Theorem 3.2 into three parts:(i) In Section 4.1, we prove the existence result for (3.2).(ii) In Section 4.2, we prove the uniqueness result for (3.2).(iii) In Section 4.3, we show that the solution of (3.2) solves (3.3).

Let us begin by recalling the classical Moreau-Yosida approximation theory. For everyλ > 0, we define the Moreau-Yosida approximation

(4.1) ϕλ(e,h) := inf(v,w)∈X

1

2λ‖(v,w)− (e,h)‖2X + ϕ(v,w).

Since ϕ : X → R is a proper, convex and l.s.c. function, Moreau’s theorem [22,Proposition 1.8, p. 162] implies that, for every λ > 0, ϕλ : X → R is a convex andFréchet differentiable function satisfying

(4.2) ϕλ(v,w) = ϕ(Jλ(v,w)) +λ

2‖Φλ(v,w)‖2X and ϕ′λ(v,w) = Φλ(v,w),

for all (v,w) ∈ X. Here, Jλ : X → X and Φλ : X → X denote, respectively, theresolvent and the Yosida approximation of the subdifferential ∂ϕ, i.e.,

(4.3) Jλ = (Id + λ∂ϕ)−1 and Φλ = λ−1(Id − Jλ

),

where Id : X → X denotes the identity operator. Since ϕ is proper, convex, andlower semicontinuous, the subdifferential ∂ϕ : X → 2X is maximal monotone (m-accretive). See, e.g, [22, Proposition 1.5, p. 157] for this well-known result. For thisreason, the Yosida approximation Φλ : X → X is maximal monotone and Lipschitz-continuous with Lipschitz constant Lλ = λ−1, and the resolvent Jλ : X → X isLipschitz-continuous with Lipschitz constant L = 1 (see [1, Theorem 3.5.9, p. 111]).


4.1. Existence. Let (f ,g) ∈ L1((0, T ),X) and (E0,H0) ∈ X. Furthermore,let λn∞n=1 be a null sequence of positive real numbers, and we consider the followingintegral equation: For every n ∈ N, find (En,Hn) ∈ C([0, T ],X) such that

(4.4)(En,Hn)(t) = Tt(E0,H0) +

∫ t

0

Tt−s((ε−1f , µ−1g)(s)− Φλn((En,Hn)(s))

)ds,

∀t ∈ [0, T ].

Since for every λ > 0 the Yosida approximation Φλ : X →X is Lipschitz-continous,a classical contraction argument [18, Theorem 1.2, p. 184] implies that, for everyn ∈ N, the integral equation (4.4) admits a unique solution (En,Hn) ∈ C([0, T ],X).Thanks to the Lipschitz continuity of Jλn

: X →X and Φλn: X →X, we have that

Jλn(En,Hn),Φλn

(En,Hn) ∈ C([0, T ],X), ∀n ∈ N.

Let us show that the sequences (En,Hn)∞n=1, Jλn(En,Hn)∞n=1, Φλn(En,Hn)∞n=1

are bounded in C([0, T ],X).Since Ttt∈R is a strongly continuous group of unitary operators on X, we may

apply the energy balance equality (Lemma 2.1) to (4.4) and obtain that

∥∥(En,Hn)(t)∥∥2

X=∥∥(E0,H0)

∥∥2

X+ 2

∫ t

0

((ε−1f , µ−1g)(s), (En,Hn)(s)

)X

− (Φλn((En,Hn)(s)), (En,Hn)(s))X ds

=∥∥(E0,H0)

∥∥2

X+ 2

∫ t

0

((ε−1f , µ−1g)(s), (En,Hn)(s)

)X

− (Φλn((En,Hn)(s))− Φλn(0, 0), (En,Hn)(s))X − (Φλn(0, 0), (En,Hn)(s))X ds,

for all t ∈ [0, T ] and n ∈ N. The monotonicity of the Yosida approximation implies

(Φλn((En,Hn)(s))− Φλn(0, 0), (En,Hn)(s))X ≥ 0, ∀s ∈ [0, T ], ∀n ∈ N,

from which it follows that∥∥(En,Hn)(t)∥∥2

X≤∥∥(E0,H0)

∥∥2

X

+ 2

∫ t

0

(‖(ε−1f , µ−1g)(s)‖X + ‖Φλn(0, 0)‖X)‖(En,Hn)(s)‖X ds, ∀t ∈ [0, T ], ∀n ∈ N.

The assumption ∂ϕ(0, 0) 6= ∅ implies that ‖Φλn(0, 0)‖X converges as n→∞ (see [22,

Theorem 1.1(c), p. 161]). For this reason, we obtain from the above inequality thatthe sequence (En,Hn)∞n=1 ⊂ C([0, T ],X) is bounded. Next, due to the Lipschitz-continuity of the resolvent Jλ : X →X with Lipschitz constant L = 1, we obtain

‖Jλn((En,Hn)(t))‖X ≤ ‖(En,Hn)(t)‖X + ‖Jλn(0, 0)‖X , ∀t ∈ [0, T ], ∀n ∈ N.

Thus, since limn→∞ ‖Jλn(0, 0)‖X = 0 (due to ∂ϕ(0, 0) 6= ∅), the above inequal-ity together with the boundedness of (En,Hn)∞n=1 ⊂ C([0, T ],X) implies thatJλn

(En,Hn)∞n=1 ⊂ C([0, T ],X) is bounded. Furthermore, by the definition of theresolvent and the Yosida approximation (4.3), we have that (see [1, Theorem 3.5.9]):

(4.5) Φλn((En,Hn)(s)) ∈ ∂ϕ (Jλn

((En,Hn)(s))) , ∀s ∈ [0, T ], ∀n ∈ N.

Thus, by Assumption 3.1 and the boundedness of Jλn(En,Hn)∞n=1 ⊂ C([0, T ],X),

(4.5) implies that Φλn(En,Hn)∞n=1 ⊂ C([0, T ],X) is bounded. In conclusion, the

IRWIN YOUSEPT 7

three sequences (En,Hn)∞n=1, Jλn(En,Hn)∞n=1, Φλn

(En,Hn)∞n=1 are boundedin C([0, T ],X). Therefore, we can select a subsequence of λn∞n=1, which we denoteagain by λn∞n=1, such that

(En,Hn) (E,H) weakly star in L∞((0, T ),X) as n→∞,(4.6)

Φλn(En,Hn) (E, H) weakly star in L∞((0, T ),X) as n→∞,(4.7)

(Id − Jλn)(En,Hn)→ 0 strongly in C([0, T ],X) as n→∞,(4.8)

Jλn(En,Hn) (E,H) weakly star in L∞((0, T ),X) as n→∞,(4.9)

for some (E,H)(E, H) ∈ L∞((0, T ),X). We note that (4.8) follows from the bound-edness of Φλn

(En,Hn)∞n=1 ⊂ C([0, T ],X) and the definition of the Yosida approx-imation Φλ = λ−1

(Id − Jλ

). Moreover, (4.9) follows from (4.8) and (4.6).

Passing to the limit n→∞ in (4.4), we obtain from (4.6) and (4.7) that

(4.10) (E,H)(t) = Tt(E0,H0) +

∫ t

0

Tt−s(

(ε−1f , µ−1g)(s)− (E, H)(s))ds, ∀t ∈ [0, T ].

and for every t ∈ [0, T ]

(4.11) (En,Hn)(t) (E,H)(t) weakly in X as n→∞.

Now, employing the classical result by Ball [3], the solution of (4.10) satisfies

(4.12)

d

dt((E,H)(t), (v,w))X − ((E,H)(t),A∗(v,w))X = ((ε−1f , µ−1g)(t)

− (E, H)(t), (v,w))X ,

for a.e. t ∈ (0, T ) and all (v,w) ∈ D(A∗),(E,H)(0) = (E0,H0),

and, for every (v,w) ∈ D(A∗), the mapping t 7→ ((E,H)(t), (v,w))X is absolutelycontinuous from [0, T ] to R.

Since the Maxwell operator A is skew adjoint, i.e., D(A∗) = D(A) = H0(curl)×H(curl) and A∗ = −A (see (2.4) on p. 2 for its definition), we see that (4.12) isequivalent to

(4.13)

d

dt

∫Ω


∫Ω


+

∫Ω


∫Ω

f(t) · v + g(t) ·w dx,


(E,H)(0) = (E0,H0).

For this reason, if we can prove that

(4.14) (E, H)(t) ∈ ∂ϕ((E,H)(t)) for a.e. t ∈ (0, T ),

then we see that (E,H) ∈ C([0, T ],X) is a solution to (3.2). To this aim, we introducethe set B ⊂ L2((0, T ),X)× L2((0, T ),X) defined as follows:

(4.15) ((e,h), (e, h)) ∈ B ⇔ (e, h)(t) ∈ ∂ϕ((e,h)(t)) for a.e. t ∈ (0, T ).


By definition, we see that (4.14) is nothing but

(4.16) ((E,H), (E, H)) ∈ B.

Therefore, we have to show that (4.16) is satisfied.Since ∂ϕ : X → 2X is monotone, the set B is monotone, i.e.,

((e, h)−(v, w), (e,h)−(v,w))L2((0,T ),X) ≥ 0, ∀((e,h), (e, h)), ((v,w), (v, w)) ∈ B.

Let us now show that B is maximal monotone (m-accretive), i.e., we have to showthat, for every λ > 0, it holds that

(4.17) Range(Id + λB) := (e,h) + λ(e, h) | ((e,h), (e, h)) ∈ B = L2((0, T ),X).

Let λ > 0. To show (4.17), we take an arbitrarily fixed function (v,w) ∈ L2((0, T ),X)and define

(e,h)(t) := Jλ((v,w)(t)) = (Id + λ∂ϕ)−1(v,w)(t) for a.e. t ∈ (0, T ).

According to the definition (4.15), we see that (4.17) is valid, if (e,h) ∈ L2((0, T ),X).Indeed, due to the Lipschitz continuity of Jλ : X → X (with Lipschitz constantL = 1), we obtain that (e,h) = Jλ(v,w) is measurable and

‖(e,h)(t)‖X ≤ ‖Jλ((v,w)(t))− Jλ(0, 0)‖X + ‖Jλ(0, 0)‖X≤ ‖(v,w)(t)‖X + ‖Jλ(0, 0)‖X for a.e. t ∈ (0, T ).

Since (v,w) ∈ L2((0, T ),X), it follows therefore that (e,h) ∈ L2((0, T ),X). Inconclusion, B is maximal monotone.

Now, making use of the energy balance equality (Lemma 2.1) in (4.10) and (4.4),we obtain that

‖(E,H)(T )‖2X = ‖(E0,H0)‖2X + 2

∫ T

0

((ε−1f , µ−1g)(t)− (E, H)(t), (E,H)(t))X dt,

‖(En,Hn)(T )‖2X = ‖(E0,H0)‖2X

+ 2

∫ T

0

((ε−1f , µ−1g)(t)− Φλn((En,Hn)(t)), (En,Hn)(t))X dt, ∀n ∈ N.

Combining these two identities results in

2

∫ T

0

(Φλn((En,Hn)(t)), (En,Hn)(t))X dt = −‖(En,Hn)(T )‖2X + ‖(E0,H0)‖2X

+2

∫ T

0

((ε−1f , µ−1g)(t), (En,Hn)(t))L2(Ω) dt

= −‖(En,Hn)(T )‖2X + ‖(E,H)(T )‖2X + 2

∫ T

0

((E, H)(t), (E,H)(t))X dt

+2

∫ T

0

((ε−1f , µ−1g)(t), (En,Hn)(t)− (E,H)(t))L2(Ω) dt.

IRWIN YOUSEPT 9

Then, employing (4.8) in the above identity, we infer that

2 lim infn→∞

∫ T

0

(Φλn((En,Hn)(t)), Jλn

((En,Hn)(t)))X dt

= 2 lim infn→∞

∫ T

0

(Φλn((En,Hn)(t)), (En,Hn)(t))X dt

≤ 2 lim supn→∞

∫ T

0

(Φλn((En,Hn)(t)), (En,Hn)(t))X dt

≤ lim supn→∞

(− ‖(En,Hn)(T )‖2X

)+ ‖(E,H)(T )‖2X + 2

∫ T

0

((E, H)(t), (E,H)(t))X dt

+ 2 lim supn→∞

∫ T

0

((ε−1f , µ−1g)(t), (En,Hn)(t)− (E,H)(t))L2(Ω) dt

= − lim infn→∞

‖(En,Hn)(T )‖2X + ‖(E,H)(T )‖2X + 2

∫ T

0

((E, H)(t), (E,H)(t))X dt

+ 2 lim supn→∞

∫ T

0

((ε−1f , µ−1g)(t), (En,Hn)(t)− (E,H)(t))L2(Ω) dt.

It follows therefore from the weak convergence properties (4.6) and (4.11) with t = Tthat

(4.18)lim infn→∞

∫ T

0

(Φλn((En,Hn)(t)), Jλn

(En,Hn)(t))X dt

≤∫ T

0

((E, H)(t), (E,H)(t))X dt.

On the other hand, according to (4.5) and by the definition (4.15), it holds that

(4.19) (Jλn(En,Hn),Φλn(En,Hn)) ∈ B, ∀n ∈ N.

Due to the maximal monotonicity property of B, the inclusion (4.19), the weak con-vergence properties (4.7) and (4.9), we obtain from the inequality (4.18) that (4.16)is satisfied. This is due to the well-known monotonicity trick (see [22, Proposition 1.6p. 159]). In conclusion, (E,H) ∈ C([0, T ],X) satisfies (4.13)-(4.14), and hence it is asolution of (3.2).

4.2. Uniqueness. Let (f ,g) ∈ L1((0, T ),X) and (E0,H0) ∈ X. Supposethat (E(1),H(1)), (E(2),H(2)) ∈ C([0, T ],X) satisfy (3.2) and, for every (v,w) ∈H0(curl) ×H(curl), the mappings t 7→ ((E(j),H(j))(t), (v,w))X , j = 1, 2 are ab-solutely continuous from [0, T ] to R. By definition, the difference (e,h) := (E(1) −E(2),H(1) −H(2)) satisfies (e,h)(0) = 0 and

(4.20)

d

dt

∫Ω

εe(t) · v + µh(t) ·w dx+

∫Ω

e(t) · curlw − h(t) · curlv dx

=

∫Ω

ε(E

(2)(t)− E

(1)(t))· v + µ

(H

(2)(t)− H

(1)(t))·w dx

for a.e. in (0, T ) and all (v,w) ∈H0(curl)×H(curl),

for some (E(j), H

(j)) ∈ L∞((0, T ),X), j = 1, 2, satisfying

(4.21) (E(j), H

(j))(t) ∈ ∂ϕ((E(j),H(j))(t)) for a.e. t ∈ (0, T ).


In view of (4.20), the classical result by Ball [3] implies that (e,h) ∈ C([0, T ],X)satisfies the variation by constants formula

(e,h)(t) =

∫ t

0

Tt−s((E(2)− E

(1), H

(2)− H

(1))(s)) ds, ∀t ∈ [0, T ].

Therefore, we obtain from the energy balance equality (Lemma 2.1) together with(e,h)(0) = 0 that

‖(e,h)(t)‖2X = 2

∫ t

0

((E(2)− E

(1), H

(2)− H

(1))(s), (e,h)(s))Xds

= −2

∫ t

0

((E(2)− E

(1), H

(2)− H

(1))(s), (E(2) −E(1),H(2) −H(1))(s))Xds ≤ 0,

for all t ∈ [0, T ], where we have used the monotonicity property of the subdifferential∂ϕ to get the above inequality. In conclusion, (E(1),H(1)) = (E(2),H(2)). Then,applying (e,h) = 0 to (4.20), we obtain for a.e. t ∈ (0, T ) and all (v,w) ∈H0(curl)×H(curl) that∫

Ω

ε(E

(2)(t)− E

(1)(t))· v + µ

(H

(2)(t)− H

(1)(t))·w dx = 0,

from which it follows that (E(1), H

(1)) = (E

(2), H

(2)).

4.3. Final Step: (3.2) ⇒ (3.3). Let (f ,g) ∈ L1((0, T ),X) and (E0,H0) ∈ X.Further, let (E,H) ∈ C([0, T ],X) denote the unique solution to (3.2), as we havealready proved in the previous steps. In other words, for every (v,w) ∈H0(curl)×H(curl), the mapping t 7→ ((E,H)(t), (v,w))X is absolutely continuous from [0, T ]to R, and it holds that

(4.22)

d

dt

∫Ω


∫Ω


+

∫Ω


∫Ω

f(t) · v + g(t) ·w dx,


(E,H)(0) = (E0,H0),

with a unique (E, H) ∈ L∞((0, T ),X) satisfying (E, H)(t) ∈ ∂ϕ((E,H)(t)) for a.e.t ∈ (0, T ). Using again the classical result by Ball [3], the variational formulation(4.22) is equivalent to the variation of constants formula:

(4.23)(E,H)(t) = Tt(E0,H0) +

∫ t

0

Tt−s(

(ε−1f , µ−1g)(s)− (E, H)(t))ds,

∀t ∈ [0, T ].

Then, applying the energy balance equality (Lemma 2.1) to (4.23), we obtain that

‖(E,H)(t)‖2X = ‖(E0,H0)‖2X + 2

∫ t

0

((ε−1f , µ−1g)(t)− (E, H)(t), (E,H)(t))X dt,

∀t ∈ [0, T ],

IRWIN YOUSEPT 11

from which it follows that the mapping t 7→ ‖(E,H)(t)‖2X is absolutely continuousfrom [0, T ] to R with

(4.24)1

2

d

dt‖(E,H)(t)‖2X = ((ε−1f , µ−1g)(t)− (E, H)(t), (E,H)(t))X

for a.e. t ∈ (0, T ).

Since (E, H)(t) ∈ ∂ϕ((E,H)(t)) holds for a.e. t ∈ (0, T ), we obtain from the defini-tion of the subdifferential that

(4.25)−((E, H)(t), (E,H)(t))X ≤ −((E, H)(t), (v,w))X + ϕ(v,w)− ϕ((E,H)(t)),

for a.e. t ∈ (0, T ) and all (v,w) ∈X.

Then, applying (4.25) to (4.24) yields that

(4.26)

− d

dt

∫Ω

εE(t) · 1

2E(t) + µH(t) · 1

2H(t) dx+

∫Ω

f(t) ·E(t) + g(t) ·H(t) dx

−∫

Ω

εE(t) · v + µH(t) ·w dx+ ϕ(v,w)− ϕ((E,H)(t)) ≥ 0,

for a.e. t ∈ (0, T ) and all (v,w) ∈X.

Finally, adding (4.26) to the left hand side of the variational equality in (4.22), wesee that (E,H) satisfies the Maxwell variational inequality (3.3).

5. Proof of Theorem 3.3. In the following, let (f ,g) ∈ C0,1([0, T ],X) and(E0,H0) ∈H0(curl)×H(curl).

5.1. Uniqueness. Suppose that (E(j),H(j)) ∈ L∞((0, T ),H0(curl)×H(curl))∩W 1,∞((0, T ),X), j = 1, 2, are solutions to (VI). Setting the test function (v,w) =

(E(2),H(2))(t) in the variational inequality for (E(1),H(1)) and the test function(v,w) = (E(1),H(1))(t) in the variational inequality for (E(2),H(2)), and then addingthe resulting inequalities, we obtain for the difference (e,h) := (E(1) −E(2),H(1) −H(2)) that

−∫

Ω

εd

dte(t) · e(t) + µ

d

dth(t) · h(t) dx−

∫Ω

curlH(1)(t) ·E(2)(t) dx

+

∫Ω

curlE(1)(t) ·H(2)(t) dx−∫

Ω

curlH(2)(t) ·E(1)(t) dx

+

∫Ω

curlE(2)(t) ·H(1)(t) dx ≥ 0, for a.e. t ∈ (0, T ).

On the other hand, we know that E(j) ∈H0(curl) and H(j) ∈H(curl) for j = 1, 2such that (2.1) implies

−∫

Ω

curlH(1)(t) ·E(2)(t) dx+

∫Ω

curlE(1)(t) ·H(2)(t) dx

−∫

Ω

curlH(2)(t) ·E(1)(t) dx+

∫Ω

curlE(2)(t) ·H(1)(t) dx = 0.

It follows therefore that

0 ≥∫

Ω

εd

dte(t) · e(t) + µ

d

dth(t) · h(t) dx =

1

2

d

dt‖(e,h)(t)‖2X a.e. in (0, T ).

Integrating this inequality yields that 0 ≥ ‖(e,h)(t)‖2X−‖(e,h)(0)‖2X = ‖(e,h)(t)‖2Xfor all t ∈ [0, T ].


5.2. Existence. Let λn∞n=1 be a null sequence of positive real numbers, andwe consider again the integral equation (4.4). Thanks to the regularity (f ,g) ∈C0,1([0, T ],X) and (E0,H0) ∈ D(A), we may apply a well-known result from thesemilinear semigroup theory (see [18, Theorem 1.6, p. 189] and [10, Corollary 7.6, p.440]) and obtain that (4.4) admits a unique solution (En,Hn) ∈ C([0, T ], D(A)) ∩C1([0, T ],X) satisfying (En,Hn)(0) = (E0,H0) and

(5.1)d

dt(En,Hn)(t)−A(En,Hn)(t) = (ε−1f , µ−1g)(t)−Φλn((En,Hn)(t)), ∀t ∈ [0, T ].

Moreover, the sequence (En,Hn)∞n=1 is bounded in C([0, T ], D(A))∩C1([0, T ],X).This boundedness result will be shown in Section 5.3. Therefore, as readily proven inSection 4.1 and due to the boundedness of (En,Hn) ⊂ C([0, T ], D(A))∩C1([0, T ],X),we can select a subsequence of λn∞n=1 in (4.4), denoted again by the sequence itself,such that

(En,Hn) (E,H) weakly star in L∞((0, T ),X) as n→∞,

Φλn(En,Hn) (E, H) weakly star in L∞((0, T ),X) as n→∞,

where (E,H) ∈ L∞((0, T ), D(A)) ∩W 1,∞((0, T ),X) and (E, H) ∈ L∞((0, T ),X)satisfy (3.2). Employing the higher regularity property (E,H) ∈ L∞((0, T ), D(A))∩W 1,∞((0, T ),X) in (3.2) and D(A) = X, we obtain that

(5.2)

∫Ω

εd

dtE(t) · v + µ

d

dtH(t) ·w dx+

∫Ω

curlE(t) ·w − curlH(t) · v dx

+

∫Ω


∫Ω

f(t) · v + g(t) ·w dx,

for a.e. t ∈ (0, T ) and all (v,w) ∈X,

(E(t), H(t)) ∈ ∂ϕ((E,H)(t)) for a.e. t ∈ (0, T ),

(E,H)(0) = (E0,H0).

On the other hand, as readily proven in (4.26), it holds for a.e. t ∈ (0, T ) and all(v,w) ∈X that

− d

dt

∫Ω

εE(t) · 1

2E(t) + µH(t) · 1

2H(t) dx+

∫Ω

f(t) ·E(t) + g(t) ·H(t) dx

−∫

Ω

εE(t) · v + µH(t) ·w dx+ ϕ(v,w)− ϕ((E,H)(t)) ≥ 0,

from which together with the regularity property (E,H) ∈W 1,∞((0, T ),X) it followsthat

(5.3)−∫

Ω

εd

dtE(t) ·E(t) + µ

d

dtH(t) ·H(t) dx+

∫Ω

f(t) ·E(t) + g(t) ·H(t) dx

−∫

Ω

εE(t) · v + µH(t) ·w dx+ ϕ(v,w)− ϕ((E,H)(t)) ≥ 0.

Adding (5.3) to the left hand side of the variational equality in (5.2), we concludethat (E,H) satisfies (VI).

IRWIN YOUSEPT 13

5.3. Boundedness Result for (4.4). Let t ∈ (0, T ) and n ∈ N. According to(4.4), for every h ∈ (0, T − t), it holds that

(En,Hn)(t+ h)

= Tt+h(E0,H0) +

∫ t+h

0

Tt+h−s((ε−1f , µ−1g)(s)− Φλn((En,Hn)(s))

)ds

= Tt+h(E0,H0) +

∫ h

0

Tt+h−s((ε−1f , µ−1g)(s)− Φλn

((En,Hn)(s)))ds

+

∫ t+h

h

Tt+h−s((ε−1f , µ−1g)(s)− Φλn

((En,Hn)(s)))ds,

= Tt(Th(E0,H0) +

∫ h

0

Th−s((ε−1f , µ−1g)(s)− Φλn((En,Hn)(s))

)ds

)+

∫ t

0

Tt−s((ε−1f , µ−1g)(s+ h)− Φλn

((En,Hn)(s+ h)))ds.

Subtracting (4.4) from the above expression, it follows that

(En,Hn)(t+ h)− (En,Hn)(t)

h

= Tt(Th(E0,H0)− (E0,H0)

h+

1

h

∫ h

0

Th−s((ε−1f , µ−1g)(s)

− Φλn((En,Hn)(s))) ds

)+

1

h

∫ t

0

Tt−s(

(ε−1f , µ−1g)(s+ h)− (ε−1f , µ−1g)(s)

− Φλn((En,Hn)(s+ h)) + Φλn

((En,Hn)(s))

)ds.

Then, applying the energy balance equality (Lemma 2.1) to the above variation ofconstants formula implies∥∥∥∥ (En,Hn)(t+ h)− (En,Hn)(t)

h

∥∥∥∥2

X

=

∥∥∥∥Th(E0,H0)− (E0,H0)

h+

1

h

∫ h

0

Th−s((ε−1f , µ−1g)(s)− Φλn((En,Hn)(s))

)ds

∥∥∥∥2

X

+2

∫ t

0

[((ε−1f , µ−1g)(s+ h)− (ε−1f , µ−1g)(s)

h,

(En,Hn)(s+ h)− (En,Hn)(s)

h

)X

− 1

h2(Φλn((En,Hn)(s+ h))− Φλn((En,Hn)(s)), (En,Hn)(s+ h)− (En,Hn)(s))X

]ds

=: In(h) + IIn(h).

Since Th−s : X →X is unitary, the first term In(h) can be estimated as follows:

In(h) ≤[∥∥∥∥Th(E0,H0)− (E0,H0)

h

∥∥∥∥X

+1

h

h∫0

‖(ε−1f , µ−1g)(s)− Φλn((En,Hn)(s))‖X ds

]2

≤[∥∥∥∥Th(E0,H0)− (E0,H0)

h

∥∥∥∥X

+ ‖(ε−1f , µ−1g)‖C([0,T ],X) + ‖Φλn(En,Hn)‖C([0,T ],X)

]2

.

Therefore, since Φλn(En,Hn)∞n=1 ⊂ C([0, T ],X) is bounded (see Section 4.1) and

(E0,H0) ∈ D(A), there exists a constant c > 0, independent of n, t and h, such that

(5.4) In(h) ≤ c, ∀h ∈ (0, T − t).


On the other hand, thanks to the monotonicity property of the Yosida approximation,the second term IIn(h) can be estimated as follows:

IIn(h)

≤ 2

t∫0

((ε−1f , µ−1g)(s+ h)− (ε−1f , µ−1g)(s)

h,

(En,Hn)(s+ h)− (En,Hn)(s)

h

)X

ds

≤ 2T

∥∥∥∥(ε−1f , µ−1g)

∥∥∥∥C0,1([0,T ],X)

∥∥∥∥ ddt (En,Hn)

∥∥∥∥C([0,T ],X)

≤ 2T 2

∥∥∥∥(ε−1f , µ−1g)

∥∥∥∥2

C0,1([0,T ],X)

+1

2


∥∥∥∥2

C([0,T ],X)

, ∀h ∈ (0, T − t).

In conclusion, it holds for all h ∈ (0, T − t) that∥∥∥∥ (En,Hn)(t+ h)− (En,Hn)(t)

h

∥∥∥∥2

X

≤ c+ 2T 2

∥∥∥∥(ε−1f , µ−1g)

∥∥∥∥2

C0,1([0,T ],X)

+1

2


∥∥∥∥2

C([0,T ],X)

.

Then, passing to the limit h→ 0,∥∥∥∥ ddt (En,Hn)(t)

∥∥∥∥2

X

≤c+ 2T 2

∥∥∥∥(ε−1f , µ−1g)

∥∥∥∥2

C0,1([0,T ],X)

+1

2


∥∥∥∥2

C([0,T ],X)

.

As t ∈ (0, T ) and n ∈ N were chosen arbitrarily, it follows therefore that

1

2


∥∥∥∥2

C([0,T ],X)

≤c+ 2T 2

∥∥∥∥(ε−1f , µ−1g)

∥∥∥∥2

C0,1([0,T ],X)

, ∀n ∈ N,

and consequently ddt (En,Hn)∞n=1 ⊂ C([0, T ],X) is bounded. Thus, from (5.1) andthe boundedness of Φλn(En,Hn)∞n=1 ⊂ C([0, T ],X), we obtain that (En,Hn)∞n=1

is bounded in C([0, T ], D(A)) ∩ C1([0, T ],X).

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