lecture notes on calculus of variations and partial ... · differential equations (2s-16/17)....

LECTURE NOTES ON CALCULUS OF VARIATIONS AND PARTIAL

DIFFERENTIAL EQUATIONS (2S-16/17).

LECTURE 1-12, DRAFT (LECTURE 12: MAY 8 (MAY 9,10: STUDENTS

PRESENTATION))

MARGARIDA BAIA, DM, IST

Lecture 1.

1. Introduction

Calculus of variations is a part of mathematical analysis dedicated to the study of theextrema (maximum/minimum, “local or global”’) and critical points of functionals (a mapfrom an infinite-dimensional function space to the real numbers; see e.g Brezis [10], Kreyszig[29], Rudin [33]). One example of such a functional could be

I(u) =

∫ b

a

√1 + [u′(x)]2 dx

defined for all u = u(x) continuously differentiable functions defined on the interval [a, b] ⊂ R,whose value for each of such function u turns out to be the length of its graph. One could beinterested, for instance, on finding the minimum of I restricted to some sub-class of functions.

We will see1 that studying extrema of functionals is a generalization of the problem ofstudying extrema of functions of several variables in calculus (see e.g Apostol [4, 5]). Thevariables will themselves be functions and we will be seeking to study the extrema of “func-tions of functions”. These functionals are generally defined by definite integrals involvingfunctions (and their derivatives) often characterized by boundary conditions and/or otherconstraints and smoothness requirements, depending on the problem under study.

Calculus of variations (whose name is due to the technique of variation that is employedto obtain certain necessary conditions for the existence of extreme values as we will studyin Subsection 2.4.1) has roots in many areas from geometry to optimization to mechanics.We refer to Subsection 1.5 for historical surveys on this subject, whose enormous growth(rigorous foundation and understanding) turned into a satisfactory theory only around thefirst half of the 20th century and forces any attempt to completely describe it as a wholetheory most likely to an impossible mission.

As far as applications, we all have already had the opportunity to observe Nature’s andHuman’s propensity to “minimize” efforts and “maximize” benefits. Indeed in physics varia-tional principles involving these kind of functionals or energies (as for instance Fermat’s (orleast time) principle, least-action principle, the law of maximal entropy, Hamilton’s princi-ple) form one of the most wide-ranging means of formulating mathematical models governing

1Most of the material presented on this series of notes is based on given references; any typo/mistakedetection are welcome to be sent to me by email

1

2 MARGARIDA BAIA, DM, IST

the equilibrium configurations of physical systems. In economics, management and finance,problems as the minimization of the cost, maximization of the profit or minimization of theinvestment risk are of the fundamental interest. As an example we mention other fields:Aeronautics (maximization of the lift of an airplane wing; optimum flight profiles from akinetic energy point of view and energy/fuel consumption economy); Mechanical engineering(maximization of the strength of a column, a dam or an arch); Electrical engineering (bestelectronic filter and reflector antennas design); Sport and related equipment design (mini-mization of the air resistance on a bicycle helmet, optimum shape of a ski, optimum shapeof a boat hull); Computer Vision (image segmentation, image morphing and image denoisingmodels are based on minimization problems).

Calculus of Variations is used in other areas of mathematics as, for instance, differen-tial geometry (geodesic, minimal surface and isoperimetric problems), differential equations(study of the existence of solutions for ordinary and partial differential equations, even insome situations when these cannot be found analytically, as the “three-body problem”, seeSubsection 1.5), and it gives a base for optimal control theory, with many applications inthe problems mentioned above. We refer to Subsection 1.5 for further reading regardingapplications.

1.1. Notation: This section contains the general notation (unless otherwise specified) usedduring the course.

- A typical point in RN (n-dimensional real Euclidean space supplied with a Cartesiansystem of coordinates with origin at ORN ) is x = (x1, ..., xN ).

- Euclidean norm in RN : ||x||2 =√∑N

i=1 xi2 (most of the times denoted by || · ||).

- Ω ⊂ RN represents an open set.- Given F : R3 → R3, F = (F1, F2, F3), ∇ · F ≡ div F := ∂F1

∂x1+ ∂F2

∂x2+ ∂F3

∂x3and

∇× F = curl(F ) := (∂F3∂x2− ∂F2

∂x3, ∂F1∂x3− ∂F3

∂x1, ∂F2∂x1− ∂F1

∂x2).

- C(Ω) and C(Ω;Rd) stand, respectively, for the set of continuous functions u : Ω→ Ror u : Ω→ Rd, accordingly.

- C(Ω) is the set of continuous functions u : Ω → R, which can be continuouslyextended to Ω

- A vector α = (α1, ..., αN ) with αi ≥ 0 is called a multi-index of order |α| :=∑N

i=1 αi.

- Ck(Ω) is the set of functions u : Ω → R which have all partial derivatives, Dαu :=∂|α|u

∂xα11 ...∂x

αNN

, with |α| ≤ k, continuous.

- Ck(Ω) is the subset of Ck(Ω) of those functions whose derivatives up to the order kcan be extended continuously to Ω.

- C∞(Ω) = ∩∞k=0Ck(Ω) (infinitely differentiable functions).

- C∞c (Ω) stands for those functions in C∞(Ω) with compact support, i.e, C∞c (Ω) =u ∈ C∞(Ω), u(x) = 0 ∀x ∈ Ω \K, K ⊂ Ω compact set.

- Obvious adaptation for C(Ω;Rd), Ck(Ω;Rd), Ck(Ω;Rd), C∞(Ω;Rd), C∞c (Ω;Rd).- We write A = (ai,j)i=1,...,d; j=1,...,N to mean a d×N matrix (or, by identification, an

element in Rd×N ) with (i, j)th entry ai,j ∈ R.- tr(A) and cof(A) stand for the trace and cofactor matrix of A ∈ Rd×N .

- If A = (ai,j)i=1,...,d j=1,...,N ∈ Rd×N then we denote |A| =(∑d

i=1

∑Nj=1 a

2i,j

)1/2

LECTURE NOTES ON CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS (2S-16/17).LECTURE 1-12, DRAFT (LECTURE 12: MAY 8 (MAY 9,10: STUDENTS PRESENTATION))3

- Du ≡ ∇u =(∂ui

∂xj

)i=1,...,d; j=1,...,N

∈ Rd×N for u : Ω → Rd, u = (u1, ..., ud) (gradient

vector if K = 1).

- D2u =(

∂2u∂xi∂xj

)i,j=1,...,N

(Hessian matrix).

- ∆u =∑N

i=1∂2u∂xi2

, u : Ω→ R.- a.e (almost everywhere with respect to the Lebesgue measure in RN .)- ∞ means +∞

1.2. Model variational problem and connection to the study of PDE.

In this course, we will focus mainly on minimization problems of the form

(1.1) infu∈A

I(u)

for integral functionals I : X → R defined by

I(u) =

∫Ωf(x, u(x),∇u(x)) dx

(typically refereed as variational integrals2), where Ω ⊂ RN is an open and bounded set, theintegrand f : Ω×Rd×Rd×N → R, f = f(x, u, ξ) (e.g a density; usually called the Lagrangianof I), is a continuous3 function, X is a space of functions u : Ω→ Rd and A (admissible classof functions) consists of functions u ∈ X possibly satisfying suitable boundary conditionsand/or further constraints.

We want to minimize the integral I(u) among all functions u ∈ A. We will call m :=infu∈A I(u) the minimal value that such an integral can take. To find a minimizer of (1.1)means to discover u0 ∈ A such that

I(u0) ≤ I(u), ∀u ∈ A.

If this minimizer exits we will write I(u0) = minu∈A I(u) (or u0 = argminu∈A I(u)).

Contrary to what happens in the study of minimum problems involving functions of severalvariables, a major difficulty here is the choice of the space A depending on the one of X .A natural choice for this type of problem seems to be X = C1(Ω;Rd) (I is well defined) butunfortunately, as we will see, this space is not the best choice for guaranteeing the existenceof a minimizer. As a simple example, consider:

I(u) =

∫ 1

0

([u

′(x)]2 − 1

)2dx

and A = u ∈ C1[0, 1] : u(0) = 0, u(1) = 0.In general we will need to work with what are called Sobolev spaces.4

2Also refereed as action in the setting of optimal control theory. More general integrals could be considereddepending on higher order derivatives.

3More general integrands could be considered, e.g Caratheodory functions, but the analysis is harder froma technical point of view.

4See the appendix section (Section 4) for a short review on this class of spaces, if needed. Other spaces asfunctions of Bounded Variation could be considered depending on the problem to be studied.


Connection to PDE: Assuming that we are able to find a minimizer u0 ∈ A for problem(1.1) (also known as variational problem), let us give the idea behind its application to thestudy of (stationary) partial differential equations. Imagine that we want to solve (in thesame class A) one of these equations that, for simplicity, we write it in the abstract form

(1.2) L(u) = 0,

where L(·) denotes a given, possibly nonlinear differential operator and u is the unknown.For instance we could have

L(u) = ∆u if studying the heat equation : ∆u = 0.

Let us assume that in “some sense”, to be explained later, the general operator L(·) is the“derivative” of I(·). Then, as we will see (Subsection ??), to find the solutions of (1.2) willreduce to study the critical points of I(.). Consequently u0 will solve (1.2).

1.3. Some motivational examples. We start, as a motivation, with some specific examplesof minimization problems of the type (1.1). See Subsection 1.5 for other examples andreferences.

1.3.1. Dido’s isoperimetric problem.

Origins of the problem (adapted from the problem described at Virgil’s epic Aeneid): Givena rope (oxhide thread) of fixed length and a curve (part of the north African shoreline)determine the optimal path along which to place the rope so that the area (of land) enclosedby the rope and the curve is maximum.

Formulation of the problem: To determine the (smooth) curve γ on the plane of fixed lengthL such that the area enclosed by γ and another given planar curve σ is maximum.

For simplicity let us assume that σ is the line segment in R2 joining (−1, 0) and (1, 0). Let γbe such a curve parametrized by (x, u(x)), x ∈ [−1, 1] for all x ∈ (−1, 1) and u(−1) = u(1) = 0(assuming u = u(x) smooth5 enough). Recall that the area enclosed by this curve and theline segment (assuming u(x) ≥ 0) is given by∫ 1

−1u(x) dx

and that its length is given by ∫ 1

−1

√1 + [u′(x)]2 dx.

We are thus led to find

max

∫ 1

−1u(x) dx, u = u(x) ∈ A

where

A =

u = u(x), x ∈ [−1, 1], u(x) ≥ 0, u(−1) = u(1) = 0,

∫ 1

−1

√1 + (u′(x))2 dx = L, L > 2

.

5Smooth, for the moment at this point means: function in C1([−1, 1]) in order all quantities involved arewell defined.


At this point note that we always can reformulate a maximization problem for a func-tion/functional I into a minimization one by studying −I (see Subsection 1.4) . In theformalism of (1.1), and taking into account this remark, here f(x, u, ξ) = −u. The solutionof this problem turns out to be an arc of a circle (see Subsection 2.4.2).

Finally we refer that the term Dido’s problem has been used to cover the more generalproblem: among all closed curves in the plane of fixed perimeter (lenght) determine the curvethat encloses the maximum area. The solution turns out to be a circle (see Section ??).

1.3.2. Brachistochrone problem.

Problem: Given two points A and B in a vertical plane (but on different vertical lines),assign a path to a moving body M (with mass m > 0) along which the body, beginning fromA, will arrive to the point B falling by its own gravity, in the least time possible.

For simplicity assume A = (a1, b1) and B = (a2, b2) with ai, bi > 0 and a2 > a1 andb1 > b2 in a Cartesian coordinate system with gravity acting in the direction of the negativey-axis. We want to find a (smooth) curve γ parametrized by (x, u(x)), x ∈ [a1, a2], (assumingu = u(x) smooth6 enough) with u(x) < b1 for a1 < x ≤ a2, such that a particle with mass mslides from rest at A to B quickest among all such curves.

If this particle moves without friction, the law of conservation of mechanical energy guar-antees that the sum of its kinetic and potential energy remains constant along all the path.In addition if it starts from rest, this imply that for all x ∈ (a1, a2) then

1

2mv(x)2 +mgu(x) = mgb1,

where g is the (constant) gravitational acceleration on earth and v = v(x) the speed of theparticle at (x, u(x)). Thus for all x ∈ (a1, a2) we have that

v(x) =√

2g(b1 − u(x))

from where the total time of descent of the particle is

(1.3)1√2g

∫ a2

a1

√1 + [u′(x)]2

b1 − u(x)dx

We are thus led to find

min

1√2g

∫ a2

a1

√1 + [u′(x)]2

b1 − u(x)dx, u = u(x) ∈ A

where

A = u = u(x), x ∈ [a1, a2], u(x) < b1, u(a1) = b1, u(a2) = b2 .

In the formalism of (1.1) we remark that here f(x, u, ξ) = 1√2g

√1+[ξ]2

b1−u . The solution of this

problem turns out to be a portion of a cycloid7 (brachistochrone) (see Subsection 2.4.1.1.3).

6As before, smooth here means: function in C1([a1, b1])7A cycloid is the elongated arch that traces the path of a fixed point on a circle as the circle rolls along

a straight line in two-dimensions. In this case this line would be the horizontal line y = b1 and the circlewill roll along its bottom. The parametric form of this curve would be x(t) = a1 + k(t − sin(t)) and u(t) =


1.3.3. Minimization of the Dirichlet integral.

Problem: To find

inf

∫Ω|∇u|2 dx : u : Ω→ Rd, u = u0 on ∂Ω

Here f(x, u, ξ) = |ξ|2. The integral

∫Ω |∇u|

2 dx is called the Dirichlet integral and it appearsin several applications as:

7→ Linear elasticity8 Let us assume, for simplicity, the one-dimensional case, and let usconsider an elastic string which, at rest, is described by the segment [−1, 1] ⊂ R onthe x-axis. If we load the string and denote by u = u(x) its vertical displacement(assumed smooth enough9), then, according to the simplest model of linear elasticity(Hooke’s law), the potential elastic energy of the string is given by

k

2

∫ 1

−1|u′

(x)|2 dx

where k is a positive constant characteristic of the material from which the springis made. Assuming that the string is fixed at its boundary points and that the loadis uniformly distributed, the shape of the spring will be minimizing the total energyof the system. Let b denote a positive constant giving the uniform load distribution.We are led then to minimize

min

∫ 1

−1

[k

2|u′

(x)|2 + b u(x)

]dx, u = u(x) ∈ A

where

A = u = u(x), x ∈ [−1, 1], u(−1) = 0, u(1) = 0 .

Here f(x, u, ξ) = k2ξ

2 + b u.

7→ Study of the Laplace’s equation 10 and more generally Poisson’s equation11 see Sub-section ??. These equations appear in many contexts:

b1 − k(1 − cos(t)), t ∈ [0, T ], where the constants T and k are determined by the condition x(T ) = a2 andu(T ) = b2.

8Elasticity theory is one of the most important theories of continuum mechanics (see Subsection 1.5 forsome references in this topic). The main physical characteristic of a purely elastic material is that the energystored in the body at a given instant (a scalar quantity, often called strain energy) depends only upon theshape of the body at that instant. Consequently, returning the body to its initial shape recovers any changein energy (absence of dissipation). Apart form the strain energy, also the stress (a measure of force perunit area) and the elasticity (resistance to changes in shape) depend upon the current shape of the body.For example, metals will soften and polymers may stiffen as they are deformed to levels approaching failure.Linear elastic constitutive relations model elastic behavior of a material that is subjected to very small strains(strain: amount of deformation an object experiences compared to its original size and shape) and have linearrelations between strain and stress.

9Smooth here means again: function in C1([−1, 1])10∆u = 0: the prototype of an elliptic partial differential equation; many of its qualitative properties are

shared by more general elliptic PDE’s.11∆u = f non-homogeneous version of Laplace’s equation


(a) Potential theory e.g. in the Newtonian theory of gravity, electrostatics12, heatflow, and potential flows in fluid mechanics.

(b) Riemannian geometry e.g. the Laplace-Beltrami operator.(c) Stochastic processes e.g. the stationary Kolmogorov equation for Brownian mo-

tion.(d) Complex analysis e.g. the real and imaginary parts of an analytic function of a

single complex variable are harmonic.

1.3.4. Problems in Hyperelasticity; see also [32].

Consider a continuous body which occupies a domain13 Ω ⊂ R3 (we refer to Ω as a referenceconfiguration of this body)

A deformation (or configuration) of this body is a map u : Ω→ R3 where u(x) denotes thedeformed position of the material point x (u(Ω): deformed configuration of the body) that isassumed to be a differentiable bijection14 and orientation-preserving15, that is,

det∇u(x) > 0, x ∈ Ω.

A hyperelastic material 16 is an elastic material for which the stress-strain relationshipfollows from the existence of a scalar valued volumetric strain energy function in the referenceconfiguration, encapsulating all information regarding the material behavior. Under theassumption that the body is homogeneous (i.e, the material response is the same at eachpoint) this material is characterized by an elastic energy of the form∫

ΩW (∇u) dx

where W : R3×3 → [0,∞) is the strain-energy (or stored-energy) density of the material. Inapplications often W is given in terms of the so-called Green-St. Venant strain tensor

E =1

2

(∇v +∇vT +∇vT∇v

)where v(x) = u(x)− x (displacement).

Properties17 of W :

(1) W (I) = 0 (undeformed state costs no energy)(2) (invariance under change of frame)

W (Rξ) = W (ξ)

12Formally the theory of electricity is similar to the theory of gravity, since the forces of interaction betweenthe charges and masses separated by a distance have similar form, but nevertheless, electricity and gravityphenomena differ greatly.

13connected open set14invertibility: to avoid interpenetration of matter15∇u: is called the deformation gradient; locally represents the volume after deformation per unit original

volume.16Most materials undergo strains that qualify as large deformation and in addition in most of them the

relationship between stress and strain is nonlinear. There are many potential strain energy functions, depend-ing on the problem/model under study. Hyperelasticity (written sometimes hyper-elasticity) is the materialmodel most suited to the analysis of elastomers.

17postulates


for all rotations R and all ξ ∈ R3×3.(3) W (ξ)→∞ as detξ ↓ 0 (infinite compression costs infinite energy)(4) W (ξ)→∞ as |ξ| → ∞ (very large deformations costs infinite energy)

In the presence of an external body force field b : Ω → R3 (e.g gravity) the total elasticenergy of the system is given by

I(u) =

∫Ω

[W (∇u)− b · u] dx

Example: Ogden materials.18 In this models W is of the form

W (ξ) =m∑i=1

aitr[(ξT ξ)

γi2

]+

l∑j=1

bjtr

[cof

[(ξT ξ)

δj2

]]+ φ(detξ), ξ ∈ R3×3

where ai > 0, γi ≥ 1, bj > 0, δj ≥ 1 (material constants) and φ : R → R ∪ ∞ is a convexfunction with φ(s) → ∞ as s ↓ 0 and φ(s) = ∞ for s ≤ 0 (W is an example of what iscalled polyconvex integrand; see Section ??). Depending on the material constants this kindof model includes: Neo-Hookean solids19 and Mooney-Rivlin materials20.

The main problem in hyper-elasticity is to minimize the energy I among functions sub-ject to appropriate boundary conditions. We will use the Direct Method of the calculus ofvariations (see Section ??).

In the setting of linearized elasticity (for homogeneous, isotropic media) the elastic energyis usually given in terms of the so-called linearized strain tensor

E =1

2

(∇v +∇vT

)as

I(v) =1

2

∫Ω

2µ|E(v)|2 +

(k − 2

3µ

)|trE(v)|2 − b · u dx

where µ > 0 and k > 0 are material constants.

1.3.5. Problems in periodic homogenization: applications within the Γ-convergencetheory.

Roughly speaking, the aim of homogenization theory is to describe the behavior of micro-scopically heterogeneous composite physical structures by means of homogeneous structureswith global characteristics equivalent to the initial ones.

In many physical situations the heterogeneities are very small in comparison with theregion in which the structure is to be studied and the heterogeneities are evenly distributed,so that they can be modeled by a periodic distribution of period a small parameter. Inpractice, one is interested in the global behavior of these structures when the heterogeneities

18model used to describe the non-linear stress-strain behaviour of complex materials such as rubbers,polymers, and biological tissue.

19that can be used for predicting the nonlinear stress-strain behavior of materials undergoing largedeformations

20model where the strain energy density function is a linear combination of two invariants of the leftCauchy-Green C = ∇v∇vT deformation tensor.


are very, very small. From the mathematical point of view, we are led to characterizing theasymptotic behavior of (systems of) ordinary or partial differential equations with oscillatingperiodic coefficients of period a small parameter ε, as ε tends to zero.

A well-known model problem in periodic homogenization, used frequently to describethermal as well as electrical or linear elasticity properties in a periodic composite mediumhas as underlying the following linear second-order partial differential equation

−div(A(xε

)∇uε

)= g on Ω.

Here Ω is the (material) domain in RN (N ≥ 1), A is a scalar or tensor-valued function withperiodic coefficients, and uε and g are scalar or vector-valued functions in some appropriatefunctional spaces. One wishes to know the asymptotic behavior of the solutions uε as ε→ 0.They converge, under appropriate hypotheses, to a solution of an “homogenized” differentialequation of the form

−div(Ahom(∇u)) = g on Ω.

Starting with the use of asymptotic expansions methods, homogenization techniques evolvedtoward more general situations through other the concept, as in particular, the notion ofΓ-convergence due to De Giorgi.

From a variational point of view, the theory of periodic homogenization rests on the studyof a family of minimum problems depending on a small parameter ε > 0

(1.4) min

∫Ωfε(x, u(x),∇u(x)) dx+

∫Ωug dx : u = ϕ on ∂Ω

,

where the functions fε (the elastic density energy) are increasingly oscillating in the firstvariable as ε tends to zero, and u (the deformation), g (the density of applied body forces)and ϕ are scalar or vector-valued functions in some Sobolev space. In the example aboveif A = (Aij) and u is a scalar function, fε(x,∇u) =

∑Aij(

xε )∇iu∇ju, where ∇i = ∂/∂xi.

More general minimum problems can be considered but in this Introduction we restrict tothis case for simplicity. The homogenization of the family of minimum problems leads to an“effective homogenized minimum problem” (not depending on ε)

(1.5) min

∫Ωfhom(x, u(x),∇u(x)) dx+

∫Ωug dx : u = ϕ on ∂Ω

such that a sequence of minimizers of (1.4) converges, as ε tends to zero, to a limit u, whichis a minimizer of (1.5). The fundamental property of De Giorgi’s notion of Γ-convergence,and its main link to the other homogenization techniques, is that, under certain growth andcompactness properties on fε and some regularity on g, it implies a sequence of minimizersto have this convergence property.

Due to the the properties of Γ-convergence, the convergence of minimizers (or almostminimizers) of (1.4) to minima of (1.5) follow from the Γ-convergence of the family

(1.6) Iε(u) =

∫Ωfε(x, u(x),∇u(x)) dx


to the homogenized functional

Ihom(u) =

∫Ωfhom(x, u(x),∇u(x)) dx.

This functional provides the macroscopic, or average description, of the periodic bodyby capturing the limiting behavior of the equilibrium states of Iεε. The effective energydensity fhom is to be determined.

Other applications of Γ-convergence may include thin-structures model derivations in elas-ticity.

1.4. Exercises/Reading Projects.

• Problems with constraints and the same type of boundary conditions as Example1.3.1:

- Use Green’s Theorem to mathematically formulate Dido’s more general problem.• Problems with the same type of boundary conditions as Example 1.3.2 or 1.3.3:

- (Geodesic problem on a plane) Formulate mathematically: Find a (smooth) curvejoining two given points in the plane, with the shortest possible length21

- (Geodesic problem on a sphere) Formulate mathematically: Find a (smooth)curve, joining two given points on a sphere, with the shortest possible length22.In general: try with a general (smooth) surface of revolution

- Derive formula (1.3).- (The problem of minimal surfaces of revolution) Formulate mathematically: Find

the (smooth) curve joining two given points in the xy−plane generates by rota-tion about the x−axis the surface of smallest area possible23.

- Related problem to the one above (“soap-film” problem): finding a surface ofleast area whose boundary is a a given plane simple24 closed curve or Jordan curve(Plateau’s problem). Formulate this problem mathematically for the simple caseof a parametric surface.

- Read the mathematical formulation25 of “Fermat’s principle” (e.g in Buttazzo,Giaquinta and Hildebrant [9]).

• Problems with different restrictions.- Read the mathematical formulation26 of “Newton problem of optimal aerody-

namic profile” (e.g in Buttazzo, Giaquinta and Hildebrant [9]).• Other related problems

- Consider an electric charge density ρ : R3 → R immovable (time independent)in three-dimensional vacuum. Let E : R3 → R3 be the electric field produced by

21It is well know that the solution would be a line segment. In general one can be interested on finding theshortest path between pairs of points on other more general surfaces (or manifolds). These paths are calledgeodesics and are obtained by minimizing its length.

22Suggestion: use spherical coordinates.23The solutions of this problem, when they exist, are catenoids (curves generated by a catenary or hanging

chain). More precisely the solutions would be of the type u(x) = λ cosh(x+µλ

)for some constants λ > 0 and

µ to be determined.24i.e that does not intersect itself25not the resolution at this point26not the resolution at this point


this charge27. It can be seen that the total electric energy of ρ is given by

I =ε0

2

∫R3

(∇ · E)φdx

where φ is an electric potential field φ : R3 → R of E (i.e such that ∇φ = E)with normalizing condition φ(0) = 0.

(i) Show that (∇ · E)φ = ∇ · (Eφ)− E · (∇φ)(ii) Use Divergence Theorem to show that I = c0

2

∫R3 |∇φ|2 dx.

- Prove some of the basic properties of any extreme problem:(i) If A ⊂ B then

minu∈A

I(u) ≥ minu∈B

I(u)

(ii) It holds that

minu∈A

I(u) = −maxu∈A

[−I(u)].

(iii) Attention:

minu∈A

[I(u) + F (u)] ≥ minu∈A

I(u) + minu∈A

F (u)

(iv) Linearity: b and a > 0 real numbers

minu∈A

[aI(u) + b] = aminu∈A

I(u) + b

(v) What happens under the composition with a monotone function?

(Extra: And if inf and sup are considered instead?)

1.5. Recommended reading.

• For historical references on Calculus of Variations see Goldstine [26], Giaquinta andHildebrant [24], Freguglia and Giaquinta [20] and also Chapter 6 in Buttazzo, Gi-aquinta and Hildebrant [9]. For a nice description of some classical examples (andreferences) see the introductory chapter in Kot [28]. Finally, for general historicalreferences on mathematics see Boyer [7, 8]• Further reading:

- Menger on the calculus of variations (based on an article by Karl Menger) (http ://www − history.mcs.st− and.ac.uk/Extras/CalculusofV ariations.html)

- (*) Calculus of Variations, I. Fonseca and G. Leoni(http : //cvgmt.sns.it/media/doc/paper/2438/PCAMCalcV arF inal.pdf)

- The Feynman Lectures on Physics Volume II (The Principle of Least Action,Chapter 19; Electrostatics, Chapter 4; Electrostatic energy, Chapter 8; MaxwellEquation, Chapter 18) (http : //www.feynmanlectures.caltech.edu/) See also[19].

27For simplicity we ignore units here and we assume functions to be smooth enough for all deriva-tions/computations to make sense.


- (*) About elasticity theory: see Ciarlet [11] and Gurtin [27]. Some engineeringnotes:(http : //www.brown.edu/Departments/Engineering/Courses/En221/Notes/Elasticity/Elasticity.htm),(http : //www.umich.edu/ bme332/ch6consteqelasticity/bme332consteqelasticity.htm)

- (*) Introduction to periodic homogenization: first chapter of Allaire [2]- New insight into optimization and variational problems in the 17th century, E.

Stein and K. Wiechmann(http : //congress.cimne.com/femclass42/files/37SteinWiechmann.pdf)

- The isoperimetric problem, Viktor Blasjo(https : //www.maa.org/sites/default/files/pdf/uploadlibrary/22/Ford/blasjo526.pdf)

- (*) The isoperimetric inequality, R. Osserman(http : //www.ams.org/journals/bull/1978 − 84 − 06/S0002 − 9904 − 1978 −14553− 4/S0002− 9904− 1978− 14553− 4.pdf)

- (*) A survey on the Newton problem of optimal profiles, G. Buttazzo(http : //cvgmt.sns.it/media/doc/paper/1339/BUTTAZZO.PDF )

- (*) A new solution to the three-body problem, R. Montgomery(http : //www.ams.org/notices/200105/fea−montgomery.pdf)

(*) Some of the concepts appearing in this article/book are not expectable tobe understood at this point. Some of these concepts will be addressed along thiscourse development, and for this reason a rigorous reading of this article/book is onlyrecommended much later.

Lecture 2.

2. Classical (or indirect) Methods

The ideia of these methods28, in contrast with the direct method29 that we will study inSection ??, is assuming the existence of a solution for a variational problem of the form (1.1)to first derive (assuming certain a priori smoothness conditions) a system (or an equation,depending on the dimensions N and d) of second order differential equations, the Euler-Lagrange equations associated with functional I, which necessarily (but not sufficiently)have to be satisfied. Any solution of these equations would be called a critical (or stationary)point of I. As in the analogous case of studying extrema of a function of several variablesthen sufficient conditions would be needed to determine the nature of this critical point.

We restrict here to the case of scalar problems, that is, we only consider the case whenN = 1 or d = 1, and as a motivation we start by reviewing the study of extrema of a functionof several variables.

2.1. Recall: Finding extrema of functions of several variables.

Given Ω ⊂ RN an open set, let f : Ω→ R be a differentiable function.

28covering roughly from the end of 17th-century (related ideas) or beginning of the 18th century (firstdevelopments) to the end of the 19th one

29developed afterwards,


Necessary condition for f to have an extreme point: Recall that if x0 ∈ Ω is an extremeof f then Df(x0) = 0 (i.e x0 is a critical point of f). Indeed, being an extreme implies thatfor all v ∈ RN , the function

g(ε) = f(x0 + εv)

ε ∈ h : x0 + hv ∈ Ω, is such thatg′(0) = 0.

To study if x0 is indeed a local minimum or maximum, or a saddle point, apart from usingthe definition, which most of the times is quite difficult, a sufficient condition is to study theHessian of f . Assume that f ∈ C2 and that we are interested on the problem of finding

infx∈Ω

f(x).

Sufficient condition for having a minimizer: Assuming that Df(x0) = 0, if D2f(x0) is apositive definite matrix then x0 is a point of (strict) local minimum of f , that is, there existρ > 0 such that if 0 < ||x− x0|| ≤ ρ then f(x) > f(x0).

Conversely we recall that if x0 is a local minimum, that is, there exist ρ > 0 such thatif ||x − x0|| ≤ ρ then f(x) ≥ f(x0) it can be seen that D2f(x0) is a positive semi-definitematrix.

This two last properties relate “minimality” of f to ”convexity” properties of f as we willrecall below. Our interest on convex functions relies on the fact that this would also happenin our general case.

2.2. Remarks on convex analysis: part I.

Definition 2.1 (convex set). A set Ω ⊂ RN is said to be convex if for every x, y ∈ Ω andevery θ ∈ [0, 1] then θx+ (1− θ)y ∈ Ω.

Trivial examples: the empty set, a single point (singleton), RN itself, affine sets (a line,hyperplane), a half-space, the interior of a sphere or an ellipsoid, etc.

In general: The intersection of convex sets are convex but the union may not be convex(see Section 2.2.1).

Definition 2.2. (convex hull) The convex hull of a set Ω ⊂ RN , conv(Ω), is the set of allconvex combinations of points in Ω, i.e,

conv(Ω) =

k∑i=1

θixi : xi ∈ Ω, θi ≥ 0,k∑i=1

θi = 1

For examples see Section 2.2.1.

Definition 2.3. Let Ω ⊂ RN be a convex set. A function f : Ω→ R is said to be:

- convex iff(θx+ (1− θ)y) ≤ θf(x) + (1− θ)f(y)

for all x, y ∈ Ω and θ ∈ [0, 1].- strictly convex if if

f(θx+ (1− θ)y) < θf(x) + (1− θ)f(y)

for all x, y ∈ Ω, x 6= y, and θ ∈ (0, 1).


Geometrically the function f is convex if the line segment between any two points (x, f(x))and (y, f(y)) on its graph lies above or on its graph (strictly convex: the line segment staysstrictly above the graph).

Equivalently, f is convex if its epigraph (the set of points on or above the graph of thefunction) is a convex set.

A function is said to be (strictly) concave if −f is (strictly) convex.

Some one-dimensional examples:

(1) Convex functions: f(x) = ax+ b, x ∈ R (a, b ∈ R : given) (affine functions); g(x) =|x|, x ∈ R; h(x) = x2, x ∈ R, i(x) = ex, x ∈ R; k(x) = −log(x), x > 0.

(2) g(x) = −e−x, x ∈ R is not convex.

Question: which of the above functions are easy to check directly by definition (insteadas geometrically) that are indeed convex? Which ones are strictly convex? See Section 2.2.1.

Some examples of multivariate convex functions:

(1) Affine functions: f(x) = aT x + b, x ∈ RN (a ∈ RN , b ∈ R); g(x) = tr(ATx) + b,x ∈ Rd×N (A ∈ Rd×N , b ∈ R), are convex but not strictly convex.

(2) Some quadratic functions: f(x) = xTAx+ cTx+ d, x ∈ RN (A ∈ RN ×N, c, d ∈ RNgiven), is convex if and only if A is positive semi-definite matrix and is strictly convexif and only if A is positive definite matrix (Why? See Section 2.2.1)30.

(3) Any norm in RN (or Rd×N ). Why?31

Some trivial properties of convex functions:

- If f, g are convex functions and α, β > 0 then αf + βg is also convex.- The composition of an convex function with an affine one, is convex.- The pointwise supremum of a family of convex functions is a convex function.

Other properties of convex functions (see [21], [35]): continuous, differentiable and twicedifferentiable almost everywhere32.

Proposition 2.4 (Convexity along lines). Let Ω ⊂ RN be a convex set. A function f : Ω→ Ris convex if and only given x ∈ Ω and v ∈ RN the function g(t) = f(x + tv) defined on theset t ∈ R : x+ tv ∈ Ω is convex.

Proof. See Section 2.2.1

Theorem 2.5. Let Ω ⊂ RN be an open and convex set and let f : Ω→ R be a C2-function.Then the following conditions are equivalent:

i) f is convexii)

f(x) ≥ f(y) +∇f(y)T (x− y)

for all x, y ∈ Ω (which expresses the geometrical fact that the graph of f lies above itstangent hyperplane at the point y)

iii) D2f is positive semi-definite in Ω.

30Maybe: other criteria are needed31Recall the basic properties of a norm32with respect to the Lebesgue measure


Proof. In class.

Remark 2.6. For i)⇔ ii) is enough to assume C1.

Remark 2.7. For strictly convex functions: f (C1 function) is strictly convex in Ω if and onlyif

f(x) > f(y) +∇f(y)T (x− y)

for all x, y ∈ Ω, x 6= y. In addition if f (C2 function) is strictly convex in Ω then D2f ispositive definite in Ω. The contrary is not true. Why? See Section 2.2.1.

Theorem 2.8. Let Ω ⊂ RN be an open and convex set and let f : Ω → R be a convexC1-function. Then any critical point x0 ∈ Ω of f is an absolute minimum of f in Ω.

Proof. In class.

Remark 2.9. To be a critical point is still necessary: e.g f(x) = ex, x ∈ R is convex but itdoes not have a minimizer in R.

Lemma 2.10. Let Ω ⊂ RN be an open and convex set and let f : Ω→ R be a strictly convexfunction. Assuming that there exists a minimizer of f in Ω, then this minimizer is unique.

Theorem 2.11 (Jensen inequality: discrete version). Let Ω ⊂ RN be a convex set and letf : Ω → R be a convex function. Given xii=1,...,K points in Ω and θii=1,...,K positive

numbers with∑k

i=1 θi = 1, then

f

(k∑i=1

θixi

)≤

k∑i=1

θif(xi)

Proof. See Section 2.2.1.

Theorem 2.12 (Jensen inequality. More general version). 33 Let Ω ⊂ RN be open andbounded and f : R→ R a C1 convex function34. Given u ∈ L1(Ω), then

f

(1

|Ω|

∫Ωu(x) dx

)≤ 1

|Ω|

∫Ωf(u(x)) dx.

Proof. In class.

33Recall (see [10, 17, 21]): Let (X,M, µ) be a measure space and let us denote by Lp(X,µ), with 1 ≤ p ≤ ∞,the usual Lebesgue spaces, that is the set (of the equivalence classes) of all µ-measurable functions u : X → Rsuch that the (Lebesgue) integral

||u||Lp(X,µ) :=

(∫X

|u|p dµ)1/p

<∞

for p <∞, or

||u||L∞(X,µ) := infC ∈ [0,∞] : |u(x)| ≤ C for µ-a.e. x ∈ X <∞.

We abbreviate Lp(X,µ) by Lp(X) when this will cause no confusion (e.g. when µ = LN , X = Ω (open set inRN ) and M = LN we will simply write Lp(Ω)).

34Still true for less regular functions f .


2.2.1. Exercises.

(1) Show that the intersection of convex sets are convex but the union may not be convex.(2) Find the convex hull of Ω = (1, 0), (1, 2), (−1, 2), (2, 4), (2, 3), (6, 1) and of S =

([2, 6]× [0, 4]) \ ([3, 4]× [0, 1])(3) Show by definition that f(x) = x2 is a strictly convex function.(4) (Convexity along lines) Let Ω ⊂ RN be a convex set. Show that a function f : Ω→ R

is convex if and only given x ∈ Ω and v ∈ RN the function g(t) = f(x + tv) definedon the set t ∈ R : x+ tv ∈ Ω is convex.

(5) Show that f(x) = xTAx+ cTx+ d, x ∈ RN (A ∈ RN ×N, c, d ∈ RN given) is convexif and only if A is positive semi-definite matrix and is strictly convex if and only if Ais positive definite matrix.

(6) Show: f (C1 function) is strictly convex in Ω if and only if

f(x) > f(y) +∇f(y)T (x− y)

for all x, y ∈ Ω, x 6= y. In addition if f (C2 function) is strictly convex in Ω then D2fis positive definite in Ω. The contrary is not true.

(7) Study the convexity of the following functions: f(x, y) = x2

y , (x, y) ∈ R2, y 6= 0 and

g(x) = log(ex1 + ...+ exN ), x ∈ RN .(8) Let Ω ⊂ RN be an open and convex set and let f : Ω → R be a strictly convex

function. Assuming that there exists a minimizer of f in Ω, prove that this minimizeris unique.

(9) (Linear least square problem) Let A ∈ RN×N such that det(A) 6= 0. Justify thatx = (ATA)−1AT b is the only absolute minimizer of

minx∈RN

||Ax− b||2

(10) Convexity of some functions is very useful to prove some important properties. Provethe so-called Young-inequality by using the fact that the exponential function is astrictly convex function in R. Recall Young-inequality:

ab ≤ ap

p+bq

q, ∀a, b ≥ 0

whenever p, q ∈ (1,∞) and 1/p+ 1/q = 1; equality holds if and only if ap = bq.(11) See that f(x) = |x|p (p > 0), x ∈ RN , is convex if and only if p ≥ 1, and strictly

convex if and only if p > 1. In particular, if N = 1, a, b ≥ 0, p ≥ 1, then by theconvexity of f (

a

2+b

2

)p≤ ap

2+bp

2,

or equivalently,

(a+ b)p ≤ 2p−1(ap + bp).

(12) Show that f(x) =√

1 + |x|2, x ∈ RN , is strictly convex.


(13) Prove the discrete version of Jensen inequality. As an application of this inequalityshow that35 given xi, i = 1, ...k, positive numbers then

x1 + ...xkn

≥ n√x1...xk

Suggestion: use the fact that − log(x), x > 0, is convex.

(14) (Extended-value function). Given f : Ω ⊂ RN → R define f : RN → R as f(x) = f(x)

if x ∈ Ω and f(x) =∞ if x ∈ RN\Ω. Show that the following properties are equivalent:i) for all x, y ∈ RN and θ ∈ [0, 1]

f(θx+ (1− θ)y) ≤ θf(x) + (1− θ)f(y)

(inequality in R ∪ ∞)ii) Ω is convex and for all x, y ∈ Ω and θ ∈ [0, 1] then

f(θx+ (1− θ)y) ≤ θf(x) + (1− θ)f(y)


Convex functions properties:

- See e.g chapter two and three in [6]. Other references: [35], [21].- A. D. Alexandrov, Almost everywhere existence of the second differential of a convex

function and some properties of convex surfaces connected with it, Leningrad StateUniv. Annals [Uchenye Zapiski] Math. Ser. 6 (1939), 3-35.

- R. Howard, Alexandrov’s theorem on the second derivatives of convex functions viaRademacher’s theorem on the fisrt derivatives of Lipschitz functions (lectures notes)(http : //people.math.sc.edu/howard/Notes/alex.pdf)

Lecture 3

2.4. Study of the minimization problem (1.1) for N, d = 1.

The case N, d = 1 reduces problem (1.1) to consider Ω = (a, b) ⊂ R and an integrandf : [a, b] × R × R → R, f = f(x, u, ξ), continuous. As for the space X, in this part, weconsider X = C1([a, b]).

As for functions of several variables we start by studying necessary conditions to have aminimizer.

2.4.1. Necessary conditions for the existence of (smooth) minimizers: fixed andfree boundary values problems. Euler-Lagrange equation.

Fixed and free boundary values problems: in these problems A could be (compare withthe examples given at the introduction and other examples in the section of exercises at theIntroduction)

A = u ∈ C1([a, b]) : u(a) = α, u(b) = β, α, β ∈ R (Dirichlet boundary conditions)

or

A = u ∈ C1([a, b]) : u(a) = α, α ∈ R

35There are many other applications of this type


or

A = u ∈ C1([a, b]) : u(b) = β, β ∈ Ror

A = C1([a, b]) (free boundary case)

2.4.1.1. First case: Dirichlet boundary conditions.

Let us consider first the case where we impose Dirichlet boundary conditions (see Examples1.3.2 and 1.3.3 and other examples in the section of exercises at the Introduction). Ourobjective is to study

(2.1) infu∈A

I(u)

where I : C1([a, b])→ R is given by

I(u) =

∫ b

af(x, u(x), u

′(x)) dx

andA = u ∈ C1([a, b]) : u(a) = α, u(b) = β, α, β ∈ R

2.4.1.1.1. First necessary condition: Weak form of the Euler-Lagrange equation(or vanishing of the first variation of I).

Throughout this part we assume f to be a C1-function36. Assume now that u0 ∈ A is asolution of problem (2.1) that is

I(u0) ≤ I(u), ∀u ∈ A.

If we consider a variation of u0:

u0 + εv, ε ∈ R,with v such that u0 + εv ∈ A then, in particular,

I(u0) ≤ I(u0 + εv).

Hence we can say thatI(u0) ≤ I(u0 + εv)

for all v ∈ C1([a, b]) with v(a) = 0, v(b) = 0. Given such a function v and defining

Φ(ε) = I(u0 + εv)

we have that Φ(0) ≤ Φ(ε) for all ε ∈ R and thus 0 is a minimum point of Φ. If φ wasdifferentiable then this would imply that

(2.2) Φ′(0) =d

dεI(u0 + εv)bε=0= 0.

36for a slightly more general case: see [9]


At this point recall:

Theorem 2.13. [Differentiation under the integral sign; see e.g Dieudonne [16]]

Given [a, b] ⊂ R and Ω ⊂ RN an open set, let f : Ω× [a, b]→ R be a continuous function.

Then g(x) =∫ ba f(x, t) dt is continuous in Ω. If, in addition, the partial derivative ∂f

∂x existsand is continuous on Ω× [a, b], then g is continuously differentiable on Ω and

g′(x) =

∫ b

a

∂f

∂x(x, t) dt.

By Theorem 2.13 the function φ is indeed differentiable and (2.2) is equivalent to∫ b

a

[∂f

∂u(x, u0(x), u0

′(x))v(x) +

∂f

∂ξ(x, u0(x), u0

′(x))v

′(x)

]dx = 0

for all v ∈ C1([a, b]) with v(a) = 0, v(b) = 0. In particular

(2.3)

∫ b

a

[∂f

∂u(x, u0(x), u0

′(x))v(x) +

∂f

∂ξ(x, u0(x), u0

′(x))v

′(x)

]dx = 0

for all v ∈ C1c ([a, b]).

The integral equality (2.3) is known as the weak form of the Euler-Lagrange equationassociated with I.

A function u0 ∈ C1(a, b) satisfying (2.3) is called a weak-critical (or weak-stationary) pointof I (or a weak solution of the Euler-Lagrange equation associated to I).

The termd

dεI(u0 + εv)bε=0

(or, here, equivalently, the left term of equality (2.3)) is called the first variation of I at u0

in the direction of v and sometimes denotes by ∂I(u0, v).

Remark 2.14. Minimizers of a functional I of the above form are not necessarily C1. Example:

The functional I(u) =

∫ 1

0

([u

′(x)]2 − 1

)2dx does not have a minimizer over A = u ∈

C1([0, 1]) : u(0) = 0, u(1) = 0. Why? (in class).

2.4.1.1.2. Stronger condition for smooth minimizers: Euler-Lagrange equation as-sociated with I.

Throughout this part we assume f to be a C2-function. To derive the Euler-Lagrangeequation we will need the first version of the so-called Lemma of the Calculus of Variationpresented below.

Lemma 2.15 (Lemma of the Calculus of Variations; d = 1 smooth version). Let Ω ⊂ RN bean open set and u ∈ C(Ω) be such that∫

Ωu(x)φ(x) dx = 0, ∀φ ∈ C∞c (Ω).

Then u = 0 on Ω.


Proof. In class.

Lemma 2.16 (Lemma of the Calculus of Variations, d = 1 more general version). LetΩ ⊂ RN be an open set and u ∈ L1

loc(Ω) be such that for any φ ∈ C∞c (Ω) it follows that∫Ωu(x)φ(x) dx = 0.

Then u = 0 a.e in Ω.

Proof. For the easier case where u ∈ L2(Ω) see Theorem 1.24 in Dacorogna [14]. For a proofin L1(Ω) where also N = 1 see Lemma 1.4 in Buttazzo, Giaquinta and Hildebrant [9]. Forthis more general statement see Corollary 3.26 in Adams [1].

We are now ready to derive the Euler-Lagrange equation associated with I.

Theorem 2.17 (Euler-Lagrange equation). Given f ∈ C2([a, b] × R × R), if problem (2.1)admits a minimizer u0 ∈ A ∩ C2([a, b]) then necessarily

(2.4)d

dx

[∂f

∂ξ(x, u0(x), u0

′(x))

]=∂f

∂u(x, u0(x), u0

′(x)) for all x ∈ [a, b]

or equivalently, by the chain rule,

∂2f

∂x∂u(x, u0(x), u0

′(x)) +

∂2f

∂u∂ξ(x, u0(x), u0

′(x))u0

′(x)(2.5)

+∂2f

∂ξ2(x, u0(x), u0

′(x))u0

′′(x) =

∂f

∂u(x, u0(x), u0

′(x)) for all x ∈ [a, b].

Proof. In class.

Remark 2.18. If ∂2f∂ξ2

(x, u0(x), u0′(x)) 6= 0 then

u0′′(x) =

∂f∂u(x, u0(x), u0

′(x))

∂2f∂x∂u(x, u0(x), u0

′(x)) + ∂2f∂u∂ξ (x, u0(x), u0

′(x))u0′(x)

.

We call equation (2.4) the Euler-Lagrange equation associated with the functional I. Notethat for the case we are analizing N = d = 1 it is a non-linear second order ODE for thefunction u0, and thus it general solution would depend on two constants that it will bedetermined by the boundary conditions in A. Note also that in general the boundary valueproblem

d

dx

[∂f

∂ξ(x, u0(x), u0

′(x))

]=∂f

∂u(x, u0(x), u0

′(x)) for all x ∈ [a, b]

u(a) = α, u(b) = β

may not have a solution, and even when it does have a solution, this solution may not beunique.

A solution of (2.4) is called a critical (or stationary) point of I (sometimes also an extremalor Lagrange curve).

Let us see obtain the Euler-Lagrange equation in the following cases:


• f(x, u, ξ) = f(ξ)• f(x, u, ξ) = f(x, ξ)• f(x, u, ξ) = f(u, ξ)

In class.

Let us remark that Theorem 2.17 is not an existence result: generally a solution of (2.4)(even C2) is not a minimizer of problem (2.1). Compare these several examples:

i) I(u) =

∫ 1

0

([u

′(x)]2 − 1

)2dx, A = u ∈ C1([0, 1]) : u(0) = 0, u(1) = 0.

ii) I(u) =

∫ 1

0e−[u

′(x)]

2

dx, A = u ∈ C1([0, 1]) : u(0) = 0, u(1) = 0.

iii) I(u) =∫ 1

0 [u′(x)− 1]2 dx, A = u ∈ C1[0, 1] : u(0) = 0, u(1) = 1.

In class.

Lemma 2.19. Given f ∈ C2([a, b]×R×R) assume that u0 ∈ A ∩C2([a, b]) is a solution ofthe Euler-Lagrange equation (2.4) then

(2.6)d

dx

[f(x, u0(x), u0

′(x))− u0

′(x)

∂f

∂ξ(x, u0(x), u0

′(x))

]=∂f

∂x(x, u0(x), u0

′(x))

for all x ∈ [a, b].

Proof. In class.

Equation (2.6) is called Du Bois-Reymond equation or second form of the Euler-Lagrangeequation.

Corollary 2.20 (Other necessary condition: Second form of the Euler-Lagrange equation).Given f ∈ C2([a, b]×R×R) assume that u0 ∈ A∩C2([a, b]) is a minimizer of problem (2.1)then (2.6) holds for all x ∈ (a, b).

Remark 2.21. In general a solution of (2.6) is not a solution of (2.4). See counter-examplein Section 2.6.1.

Remark 2.22. From the proof of Lemma 2.19 it is easy to see that if the integrand does notdepend on x, that is, f = f(u, ξ), f ∈ C2, then any non-constant solution u0 ∈ A∩C2([a, b])of Du Bois-Reymond equation (2.6) is also a solution of the Euler-Lagrange quation (2.4)(from which both necessary conditions are equivalent for non-constant solutions).

We also note that in this case (2.6) imply[f(u0(x), u0

′(x))− u0

′(x)

∂f

∂ξ(u0(x), u0

′(x))

]= c, c ∈ R.

Assuming that u0 ∈ A ∩ C2([a, b]) is a solution of the Euler-Lagrange equation (2.4) anddenoting

φ(u, ξ) := f(u, ξ)− ξ ∂f∂ξ

(u, ξ)


it follows that φ(u0(x), u0′(x)) = c, c ∈ R, for all x ∈ (a, b) (conservation law). Such a

function with this property is called a first integrand for the Euler-Lagrange equation (2.4).

Example. Application in mechanics (Notation: x = x(t) instead of u = u(x))

Let

I(x) =

∫ t2

t1

(m2|x(t)|2 − V (x(t))

)dt

be the action of a motion x = x(t), t ∈ [t1, t2] of a point mass m in a conservative force fieldF = − V’ with potential energy V . If E represent the total energy of a motion x = x(t) thenit can be seen that E(x(t), x(t)) = c, c ∈ R along the solutions of Newton’s law. See section2.6.1.

Another observation that it is worth to mention about Theorem 2.17 is that minimizers,when exist in A, are not necessarily of class C2.

Example. The the functional

I(u) =

∫ 1

−1u2(x)[2x− u′

(x)]2 dx

has a minimizer over A = u ∈ C1([−1, 1]) : u(−1) = 0, u(1) = 1 that is of class C1 but notof class C2. Indeed, it can be seen that this minimization problem has a unique minimizerin A. (In class)

In general, it is not so easy to derive directly neither existence nor uniqueness of minimizers.

In analogy to the case of functions of several variables (see Lemma 2.10 and Theorem 2.8)we have the following results.

Lemma 2.23. (uniqueness) If (u, ξ) → f(x, u, ξ) is a strictly convex function37 for all x ∈[a, b] then if problem (2.1) (for a general class A) has a minimizer this minimizer is unique.

Proof. In class.

Indeed we have the following result.

Lemma 2.24. Assume (u, ξ)→ f(x, u, ξ) is a convex function for all x ∈ [a, b].

• If u→ f(x, u, ξ) is strictly convex then if problem (2.1) (for a general class A) has aminimizer, this minimizer is unique.• If ξ → f(x, u, ξ) is strictly convex then if problem (2.1) (for a general class A) has

two minimizers their derivatives must coincide.

Remark 2.25. Depending on the boundary conditions on A the second condition in Lemma2.24 may imply or not uniqueness of minimizers. For the case under study (Dirichlet boundaryconditions) it still implies uniqueness.

Lecture 4

37thus continuous


Lemma 2.26. (sufficiency) Assume f ∈ C2 and such that (u, ξ) → f(x, u, ξ) is a convexfunction for all x ∈ [a, b]. Then any solution u0 ∈ A ∩ C2 of the Euler-Lagrange equation(2.4) is a minimizer of problem (2.1).

Example.

infu∈A

I(u)

where I(u) =∫ 1

0

√1 + |u′(x)|2 dx (curve of minimal length problem). In class.

As for functions of several variables convexity by itself is not enough to guarantee existenceof minimizers.

Example. (Weierstrass example) Problem

infu∈A

I(u)

where I(u) =∫ 1

0 x[u′(x)]2 dx A = u ∈ C1([0, 1]) : u(0) = 1, u(1) = 0 has no solution. In

class.

Our main purpose in this course is to study minimizers of I in a class of admissiblefunctions A (global minimizer in that class), however let us discuss shortly the concept oflocal minimality.

Definition 2.27. A function u0 ∈ A such that

I(u0) ≤ I(u)

for all u ∈ A with ||u− v||C1 < δ for some δ > 0 is called a weal local minimizer of I.

Definition 2.28. A function u0 ∈ A such that

I(u0) ≤ I(u)

for all u ∈ A with ||u− v||sup < δ for some δ > 0 is called a strong local minimizer of I.

Recall that for u ∈ X = C1[a, b]

||u||C1 = maxx∈[a,b]

|u(x)|+ maxx∈[a,b]

|u′(x)|

and

||u||sup = maxx∈[a,b]

|u(x)|

and that both norms are not equivalent.

Examples:

• u(x) = sin (xn2)n for n sufficiently large is “close” to v(x) = 0 in [0, π] in the ||·||sup-norm

but not in the || · ||C1-norm.

• u(x) = sin (xn)n2 for n sufficiently large is “close” to v(x) = 0 in [0, π] in the ||·||C1-norm.

In class.

Clearly a strong local minimizer is a weak local minimizer. The contrary is not true asthe following example shows.


Example.infu∈A

I(u)

where

I(u) =

∫ π

0(u

′(x))2[1− (u

′(x))2] dx

and A = u ∈ C1([0, π]) : u(0) = u(π) = 0. It can be seen that u = 0 is a weak localminimizer but not a strong local minimizer of I. In class.

Lemma 2.29. If u0 ∈ A ∩ C2 is a weak local minimizer38 of I then u0 is a solution of theEuler-Lagrange equation (2.4).


2.4.1.1.3. Some classical examples.

Example. Brachistochrone problem.

min

1√2g

∫ a2

a1

√1 + [u′(x)]2

b1 − u(x)dx, u = u(x) ∈ A

where

A = u = u(x), x ∈ [a1, a2], u(x) < b1, u(a1) = b1, u(a2) = b2 .

Example. Elastic string problem.

min

∫ 1

−1

[k

2|u′

(x)|2 + b u(x)

]dx, u = u(x) ∈ A

where

A = u = u(x), x ∈ [−1, 1], u(−1) = 0, u(1) = 0 .

Both examples discussed in class: Euler-Lagrange equation and comparison with Du Bois-Reymond equation. Solutions are indeed minimizers.

Lecture 5

2.4.1.2. Other boundary values problems and free boundary value problems.

Let us start by considering the case

A = u ∈ C1([a, b]) : u(a) = α, α ∈ R

(or analogously A = u ∈ C1([a, b]) : u(b) = β, β ∈ R)Assume u0 ∈ A is a solution of problem (2.1). Then, similarly to the previous case,

I(u0) ≤ I(u0 + εv)

for all v ∈ C1([a, b]) with v(a) = 0. Proceeding as before∫ b

a

[∂f

∂u(x, u0(x), u0

′(x))v(x) +

∂f

∂ξ(x, u0(x), u0

′(x))v

′(x)

]dx = 0

for all v ∈ C1([a, b]) with v(a) = 0.

38as in particular a strong local minimizers is


Assuming u0, f ∈ C2 and integrating by parts, it follows that∫ b

a

∂f

∂u(x, u0(x), u0

′(x))− d

dx

[∂f

∂ξ(x, u0(x), u0

′(x))

]v(x)

dx

+∂f

∂ξ(b, u0(b), u0

′(b)) = 0

for all v ∈ C1([a, b]) with v(a) = 0.

We deduce the following result.

Lemma 2.30. Assume f ∈ C2. If u0 ∈ A ∩ C2 is a solution of problem (2.1) then u0 is asolution of the Euler-Lagrange equation (2.4) and in addition:

(2.7)∂f

∂ξ(b, u0(b), u0

′(b)) = 0

Proof. In class.

Condition (2.7) is refereed as a natural boundary condition on b.

Similarly for the case A = C1([a, b]) (free boundary case) we get the following result.

Lemma 2.31. Assume f ∈ C2. If u0 ∈ A ∩ C2 is a solution of problem (2.1) then u0 is asolution of the Euler-Lagrange equation (2.4) and in addition:

(2.8)∂f

∂ξ(a, u0(a), u0

′(a)) = 0,

∂f

∂ξ(b, u0(b), u0

′(b)) = 0

Proof. In class.

Condition (2.8) is refereed as natural boundary condition on a and b

2.4.1.2.1. Extensions: integrands depending on higher order derivatives.

See section 2.6.1.

2.4.1.2.2. Necessary conditions for minimizers in C1.

To be written!

2.4.2. Necessary conditions for isoperimetric problems.

To be written!

Example.

(1) Minimize

I(u) =1

2

∫ 1

0(u

′(x))2 dx

on A = u ∈ C1([0, 1]) : u(0) = 0 = u(1),∫ 1

0 u(x) dx = 16

(2) Find the critical points for

I(u) =

∫ π

0(u

′(x))2 dx

on A = u ∈ C1([0, π]) : u(0) = 0 = u(π),∫ π

0 u2(x) dx = 1


(3) (Dido’s isoperimetric problem) Maximize

I(u) =

∫ 1

−1u(x) dx

on

A =

u ∈ C1([−1, 1]), u(x) ≥ 0, u(−1) = u(1) = 0,

∫ 1

−1

√1 + (u′(x))2 dx = L, L > 2

.

(4) Hanging cable problem. To find the minimizer of

I(u) =

∫ b

0u(x)

√1 + (u′(x))2 dx

among u ∈ C1([0, b]) with u(0) = u(b) = 0 satisfying that

J(u) =

∫ b

0

√1 + (u′(x))2 dx = L, L > b.

(1)-(3) in class; (4) see section 2.6.1.

Lecture 6

2.4.2.1. Extensions/Applications.

a) Multiple isoperimetric constraint problem

To be written.

Example: See section 2.6.1.

b) Integrands with higher order derivatives

To be written.

Example: See section 2.6.1.

Lecture 7

c) Applications to eigenvalues problems

The Sturm-Liouville problem. To be written.

2.4.3. Sufficient conditions for the existence of minimizers (a brief overview).

2.4.3.1. Second variation of the functional I.

Let us consider the general problem:

(P ) infu∈A

I(u)

where I : C1([a, b])→ R is given by

I(u) =

∫ b

af(x, u(x), u

′(x)) dx

and A ⊂ C1([a, b]) is some of the admissible class of functions considered before.


So far we have used the fact that if u0 is a minimizer (or even a weak minimizer) of (P )then for every v in some appropriate class of functions

∂I(u0, v) ≡ Φ′(0) = 0

where Φ(ε) = I(u0 + εv), ε ∈ I ⊂ R.Recall for functions f of one variable: if x0 is a local minimizer then f ′(x0) = 0 and

f ′′(x0) ≥ 0; conversely if f ′(x0) = 0 and f ′′(x0) > 0 then x0 is a local minimizer of f . The

natural ideia is to study the sign of Φ′′(0) and we expect to obtain sufficient conditions for

local minimality.

Assume f ∈ C2([a, b]× R× R). It can be seen that

Φ′′(0) =

∫ b

a

∂2f

∂u2(x, u0(x), u0

′(x))v2(x) + 2

∂2f

∂u∂ξ(x, u0(x), u0

′(x))v(x)v

′(x)

+∂2f

∂ξ2

(x, u0(x), u0

′(x))

(v′)2(x)

dx

Definition 2.32. (Second variation of I at u0 in the direction of v; whenever is well defined)

∂2I(u0, v) =

∫ b

a

∂2f

∂u2(x, u0(x), u0

′(x))v2(x) + 2

∂2f

∂u∂ξ(x, u0(x), u0

′(x))v(x)v

′(x)

+∂2f

∂ξ2

(x, u0(x), u0

′(x))

(v′)2(x)

dx

2.4.3.1.1. Properties of the second variation of I: other (higher order) necessaryconditions.

Throughout this part we restrict to the case where

A = u ∈ C1([a, b]) : u(a) = α, u(b) = β, α, β ∈ R

Proposition 2.33. Assume f ∈ C2([a, b]× R× R). If u0 ∈ A is a weak local minimizer forproblem (P ) then

∂2I(u0, v) ≥ 0

for all v ∈ C1([a, b]) with v(a) = v(b) = 0.


Remark 2.34. Note that ∂2I(u0, 0) = 0. The statement of Proposition 2.33 is equivalent tosay that v = 0 is a minimum of v → ∂2I(u0)(v) ≡ ∂2I(u0, v)

Remark 2.35. If u0 is a weak local maximizer then the analogous of Proposition 2.33 holdswith ≤ instead.

Assume now that f ∈ C3([a, b]× R× R) and u0 ∈ A ∩ C2. Integrating by parts

∫ b

a2∂2f

∂u∂ξ(x, u0(x), u0

′(x))v(x)v

′(x) dx = −

∫ b

a

d

dx

(∂2f

∂u∂ξ(x, u0(x), u0

′(x))

)v2(x)

dx


Thus, we can write

∂2I(u0, v) =

∫ b

aP (x)(v

′)2(x) +Q(x)v2(x) dx

where

P (x) =∂2f

∂ξ2

(x, u0(x), u0

′(x))

and

Q(x) =∂2f

∂u2(x, u0(x), u0

′(x))− d

dx

(∂2f

∂u∂ξ(x, u0(x), u0

′(x))

)

It can be seen that

Lemma 2.36. (See [23]) Assume f ∈ C3([a, b]×R×R) and u0 ∈ A∩C2. If ∂2I(u0, v) ≥ 0for all v ∈ C1([a, b]) with v(a) = v(b) = 0 then

P (x) ≥ 0 for all x ∈ [a, b].

Theorem 2.37. (Legendre/Legendre-Hadamard necessary condition) Assume f ∈ C3([a, b]×R× R) and that u0 ∈ A ∩ C2 is a weak local minimizer of problem (P ) then

∂2f

∂ξ2

(x, u0(x), u0

′(x))≥ 0

for all x ∈ [a, b].


Example. Problem (P ) with I(u) =∫ 1−1 x

√1 + (u′)2 dx and A = u ∈ C1([−1, 1]) : u(−1) =

0, u(1) = 1 can not have a C2-weak local minimizer (neither weak local maximizer). Why?In class.

Natural question: is strict inequality (in any of the above results) a sufficient condition fora critical point of I satisfying the boundary conditions to be a weak local minimizer? Theanswer is no, as the following examples show.

Example.

(1) Problem (P ) with I(u) =∫ 1−1x

2(u′(x))2 + x(u

′(x))3 dx and A = u ∈ C1([−1, 1]) :

u(−1) = 0 = u(1). Here u0(x) = 0 is a critical point for I, ∂2I(0, v) > 0 for allv ∈ C1([−1, 1]) with v(−1) = v(1) = 0 but u0 is not a weak local minimizer for (P ).Why? In class.


(2) Problem (P ) with I(u) =∫ 5π/4

0 (u′(x))2 − u2(x) dx and A = u ∈ C1([0, 5π/4]) :

u(0) = 0 = u(5π/4). Here u0(x) = 0 is a critical point for I, ∂2f∂ξ2

(x, 0, 0) > 0 for all

x ∈ [0, 5π/4]. but u0 is not a weak local minimizer for (P ). Why? In class.

Even if condition∂2I(u0, v) > 0

for all v ∈ C1([a, b]) with v(a) = v(b) = 0 does not imply u0 to be a weak local minimizer ofproblem (P ), to understand when this holds will help to derive a sufficient condition for weaklocal minimality. Let us try to understand under which assumptions this condition hold.

We saw that (assuming f ∈ C3([a, b]× R× R) and u ∈ A ∩ C2)

∂2I(u)(v) =

∫ b

aP (x)(v

′)2(x) +Q(x)v2(x) dx

Note that the Euler-Lagrange equation for ∂2I(u) (the so called Jacobi’s equation of thefunctional I) reads as

− d

dx(Pv

′) +Qv = 0

We could associate to this ODE the following boundary conditions:

v(a) = 0, v(c) = 0

for some c ∈ a < c ≤ b. Note that v(x) = 0 is a solution of this boundary value problem, butthis problem can have more solutions.

Definition 2.38. The point a (a 6= a) is said to be conjugate to the point a if the Jacobi’sequation of the functional I has a solution that vanishes for x = a and x = a but is notidentically zero.

Example.

(1) For I(u) =∫ 1

0 (u′(x))2 dx there are not conjugate points to 0.

(2) π is a point conjugate to 0 for I(u) =∫ 5π/4

0 (u′(x))2 − u2(x) dx

In class.

Lecture 8

Theorem 2.39. (see [23]) Let f ∈ C3([a, b]× R× R) and u ∈ C2. Assume P (x) > 0 for allx ∈ [a, b]. Then ∂2I(u, v) > 0 for all v ∈ C1([a, b]) with v(a) = v(b) = 0 if and only if [a, b]contains no points conjugate to a

Theorem 2.40. (Jacobi necessary condition) Assume f ∈ C3([a, b]× R× R) and that u0 ∈A ∩ C2 is a critical point for I. If u0 is a weak local minimizer for problem (P ) and

∂2f

∂ξ2

(x, u0(x), u0

′(x))> 0

for all x ∈ [a, b], then (a, b) contains no points conjugate to a.


Proof. See Section 2.6.1

Example. Other way to justify that u0(x) = 0 is not a weak local minimizer for problem (P )

with I(u) =∫ 5π/4

0 (u′(x))2 − u2(x) dx and A = u ∈ C1([0, 5π/4]) : u(0) = 0 = u(5π/4).

If it was, by Theorem 2.40 (why? in class) then (0, 5π/4) could not contain a conjugate pointto 0. However we saw that π is a conjugate point to 0.

Summary so far (about necessary conditions for having a weak local minimizerand assum-ing the integrand f ∈ C3): If problem (P ) has a weak local minimizer39 at u0 ∈ A ∩ C2

then:

i) u0 satisfies Euler-Lagrange equation (2.4)

ii) ∂2f∂ξ2

(x, u0(x), u0

′(x))≥ 0 for all x ∈ [a, b]

iii) If ∂2f∂ξ2

(x, u0(x), u0

′(x))6= 0 then (a, b) contains no points conjugate to a.

Theorem 2.41. (see [23]) (Sufficient condition for weak local minimizer) Assume f ∈C3([a, b]× R× R) and let u0 ∈ A ∩ C2 be such:

i) u0 is a critical point for I,

ii) ∂2f∂ξ2

(x, u0(x), u0

′(x))> 0 for all x ∈ [a, b] (the strengthened Legendre’s condition),

iii) [a, b] contains no points conjugate to a (the strengthened Jacobi’s condition).

Then u0 is a weak local minimizer for problem (P ).

Example.

I(u) =∫ 1

0 (u′(x))3 + u

′(x) dx, u(0) = 0, u(1) = 2

In class.

Remarks on sufficient conditions for strong local minimizers. To be written.

Lecture 9

2.5. Remarks on Euler-Lagrange equations for N = 1 and d > 1 (system of ODE’s).To be written.

2.5.0.1. Hamiltonian formulation: a brief overview; see also Section 2.6.1. To bewritten.

Example. (Classical mechanics) To be written.

Lecture 10

2.6. Remarks on Euler-Lagrange equations for N > 1 and d = 1 (a single PDE).To be written.

Example.

(1) Dirichlet functional: Laplace and Poisson equation. (See also Section 2.6.1)(2) Remarks on the Dirichlet principle (see also [17]).

39change ≥ for ≤ for weak local maximizer


(3) Minimal Surface equation.

In class.

Remarks on other type of boundary conditions. To be written.

2.6.1. Exercises.

(1) (Notation: x = x(t) instead of u = u(x)) Let

I(x) =

∫ t2

t1

(m2|x(t)|2 − V (x(t))

)dt

be the action of a motion x = x(t), t ∈ [t1, t2] of a point mass m in a conservativeforce field F = − V’ with potential energy V . If E(x, v) = m

2 |v|2 + V (x) represent

the total energy of a motion x = x(t) show that E(x(t), x(t)) = c, c ∈ R along thesolutions of Newton’s law.

(2) Generalize Theorem 2.17 for the case u : [a, b]→ R and I(u) =∫ ba f(x, u(x), u′(x), u′′(x)) dx

and A = u ∈ C2([a, b]) : u(a) = α1, u′(a) = α2, u(b) = β1, u

′(b) = β2.(3) Obtain the Euler-Lagrange equation associated to

I(u) = H

∫ 1

−1|u′′

(x)|2 dx

where H > 0. Discuss the existence of minimizers for

infu∈A

I(u)

where A = u ∈ C2([−1, 1]) : u(−1) = u′(1) = 0, u(1) = u′(1) = 0 (simpler modelin thin elastic beam problem; H is a positive constant depending on the material ofthe beam and u = u(x) represents the vertical deflection of the beam at a point x).

(4) Assume that f = f(ξ). Use Jensen’s inequality to give an alternative proof of the

fact that if f is convex then u0(x) = β−αb−a (x − a) + α is a minimizer for our general

minimization problem.40

(5) Let Ω ⊂ RN and u ∈ L1loc(Ω) be such that for any φ ∈ C∞c (Ω) then∫

Ωu(x)φ(x) dx = 0 with

∫Ωφ(x) dx = 0.

Show that there exists a constant K such that u = K a.e in Ω. Suggestion: Use theFundamental Lemma of the Calculus of Variations to show that u =

∫Ω u(y)f(y) dy

a.e for any f ∈ C∞c (Ω) with∫

Ω f(y) dy = 1.

(6) (DuBois-Reymond’s lemma) Let Ω = (a, b) ⊂ R and u ∈ L1loc(Ω) be such that for any

φ ∈ C∞c (Ω) then ∫Ωu(x)φ

′(x) dx = 0

then there exists a constant K such that u = K a.e in Ω. Sugestion: Use the previousexercise.

40Recall that u0 is a solution of the Euler-Lagrange equation for I


(7) Use Du Bois-Reymond’s lemma above to show that if u ∈ C1([a, b]) is a solution ofthe weak Euler-Lagrange equation associated to the functional I in problem (2.1)(with f of class C1) then there exist a constant c ∈ R such that

∂f

∂ξ(x, u0(x), u0

′(x)) = c+

∫ x

a

∂f

∂u(x, u0(x), u0

′(x)) dx.

This equation is called Du Bois-Reymond’s equation or as the Euler equation ofintegrated form. Conclude that

d

dx

[∂f

∂ξ(x, u0(x), u0

′(x))

]=∂f

∂u(x, u0(x), u0

′(x)) for all x ∈ (a, b).

Note however that this is not equivalent to equation (2.5). Why?(8) Show that if u0 ∈ A ∩ C2 is a weak local minimizer of I then u0 is a solution of the

Euler-Lagrange equation (2.4).(9) (Minimal surface of revolution problem). Consider the following minimization prob-

lem:

infAI(u)

where

I(u) =

∫ b

au(x)

√1 + (u′(x))2

and A = u ∈ C1([a, b]) : u > 0, u(a) = α, u(b) = β, α, β > 0. In case there exist aminimizer justify why it should be of class C2. Obtain the Euler-Lagrange equationassociated to I.

(10) Obtain the Euler-Lagrange equation for the following functionals:

• I(u) =∫ ba |u

′(x)|2 dx (Laplace equation in 1-D)

• I(u) = 1p

∫ ba |u

′(x)|p dx, 1 < p <∞. (p-harmonic equation in 1-D)

(11) Obtain necessary conditions for a function u to be a minimizer of

I(u) =

∫ 1

0x2(u

′(x))2 + x3(u(x))2 dx

in the class A = u ∈ C1([a, b]) : u(0) = 0. Idem: if u ∈ C2.(12) Minimize

I(u) =

∫ 2

1

1

2x2(u

′(x))2 + x2u

′(x)

dx

on A = u ∈ C1([1, 2]) : u(1) = u(2) = 0(13) Minimize

I(u) =

∫ 2

1

1

2x2(u

′(x))2 + x2u

′(x)

dx

on A = u ∈ C1([1, 2]) : u(1) = 0(14) Show that there exists no minimizer of

I(u) =

∫ 1

−1x2(u′(x))2 dx

on A = u ∈ C1([−1, 1]) : u(−1) = −1, u(1) = 1.


(15) Show that for f(u, ξ) = 12ξ

2 − u the function u0(x) = 1 is a solution of (2.6) but itdoes not satisfy (2.4).

(16) Minimize

I(u) =

∫ 2

1

u

′(x)(

1 + x2u′(x))

dx

on A = u ∈ C1([1, 2]) : u(1) = 3, u(2) = 5(17) Minimize

I(u) =

∫ a

−au(x)dx

on A = u ∈ C1([−a, a]) : u(a) = α, u(b) = β, α, β > 0,∫ a−a√

1 + (u′(x))2 dx =

L, L > 2a(18) Find the critical points of

I(u) =

∫ π

0(u

′(x))2dx

on A = u ∈ C2([0, π]) : u(0) = u(π) = 0,∫ π

0 u(x) dx = 1(19) Maximize

I(u) =

∫ 1

0u(x) dx

on A = u ∈ C1([0, 1]) : u(0) = 0 = u(1),∫ 1

0

√1 + (u′(x))2 dx = π

2 (20) (Reversibility of the isoperimetric problem) If u0 is a critical point for I − λJ (I, J :

as in the result studied in class) with λ 6= 0 show that u0 is a critical point forJ − 1/λI. Study the relationship between both constrained minimization problems(i.e minimizitation of I subject to J constant and minimizitation of J subject to Iconstant).

(21) (Application of the two previous exercises) Minimize

I(u) =

∫ 1

0

√1 + (u′(x))2 dx

on A = u ∈ C1([0, 1]) : u(0) = 0 = u(1),∫ 1

0 u(x) dx = π8

(22) Find the critical points of

I(u) =

∫ 1

0(u

′(x))2dx

under the constraints ∫ 1

0u(x) dx = 2

∫ 1

0xu(x) dx =

1

2

and the boundary conditions u(0) = u(1) = 0.(23) Find the critical points of

I(u) =

∫ 1

0(u

′′(x))2dx


under the constraint ∫ 1

0u(x) dx = 1

and the boundary conditions u(0) = u(1) = 0, u′(0) = u

′(1) = 0.

(24) Proof Proposition 2.33.(25) For the following cases check if there are any conjugate points to zero.

• I(u) =∫ b

0 (u′(x))2 + x2 dx, b > 0

• I(u) =∫ b

0 (u′(x))2 − 4u2(x) + e−x

2 dx, b 6= (k + 12)π

(26) Derive the Euler-Lagrange equations for the following cases:

(a) I(u) =

∫Ω

1

2

N∑i,j=1

ai,j∂u

∂xi

∂u

∂xj− uf

dx

(b) I(u) =

∫Ω

1

2|∇u|2 − f(u)

dx

(27) Find the natural conditions at ∂Ω for problem (P ) where I(u) =

∫Ω

1

2|∇u|2 − fu

dx.

Idem if I(u) =

∫Ω

√1 + |∇u|2 dx

(28) Consider an electric charge density ρ : R3 → R immovable (time independent) inthree-dimensional vacuum. Let E : R3 → R3 be the electric field produced by thischarge41. Applying Gauss law for electricity (∇ · E = ρ/ε0, ε0 ∈ R permittivityof vacuum), Faraday law of induction (implying ∇ × E = 0) it can been seen that(Poisson equation)

4φ = − ρε0

where φ is an electric potential field φ : R3 → R of E (i.e such that ∇φ = E). Showthat solving this equation is equivalent to minimize

φ→ ε0

2

∫R3

[|∇φ|2 + ρφ

]dx.

(29) Write the (system) Euler-Lagrange equations for N > 1 and d > 1.(30) Possible projects.

(a) To present the method of coordinate transform (Invariance of the Euler-Lagrangeequation; see e.g [36]). Examples: Brachistocrone problem, Hanging cable prob-lem, others problems where it may be convinient to use a change of variables...

(b) To study necessary conditions for admissible classes of functions whose deriva-tives have jumps (broken extremals and the Weierstrass-Erdmann corner condi-tion; see e.g [36]).

(c) Higher order necessary conditions for having a minimizer (Legendre-Hadamardnecessary condition; Jacobi necessary condition; Weierstrass condition; see e.g[23])

(d) Noether’s equation and Erdmann’s equation (conservation laws). See e.g [9]

41For simplicity we ignore units here and we assume functions to be smooth enough for all deriva-tions/computations to make sense.


(e) Other constrained problems (Holonomic constraints, Nonholonomic constraints,Inequality constrains; see e.g [23])

3. Direct methods in the Calculus of Variation

Convection: scalar case: N ≥ 1, d = 1 or N = 1, d ≥ 1; vectorial case: N > 1, d > 1.

3.1. Idea of the direct method: functions of several variables.

Recall: Weierstrass theorem and its proof.

Indeed what is needed for a minimum: to guarantee that a minimizing sequence is (up to asubsequence) convergent and the function to be lower-semicontinuous. Enough: the domainto be closed and the function coercive.

To be written.

3.2. Abstract theorems on existence of minimizers using direct methods argu-ments. To be written.

Lecture 11

3.3. Brief revision on weak convergence on Banach spaces.

To be written.

3.4. Still: Abstract theorems on existence of minimizers using direct methodsarguments on Banach spaces. To be written.

Lecture 12

3.5. Brief revision on weak convergence on Lp and W 1,p spaces.

To be written.


(1) For general reading on the Classical (or indirect) Method see [14], [9], [25], [36], [23](2) For a review on Lp-spaces see [10] or [21].(3) For a review on ODE’s see [12] or [13].(4) For an introduction to the Direct Method of the Calculus od Variations see [14] and

[17] (see also [15] for a more detailed analysis).

4. Appendix: Prerequisites

To be written...

(Note: References below are still incomplete...)


References

[1] A. R. Adams, Sobolev spaces, Academic Press, New York, 1975.[2] G. Allaire, Shape optimization by the homogenization method, Springer, New York, 2002.

[3] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford

Mathematical Monographs. Oxford: Clarendon Press, 2000.[4] T. M. Apostol, Calculus, Vol. 1, One-variable calculus, with an introduction to linear algebra, Wiley, New York,

1967.[5] T. M. Apostol, Calculus, Vol. 2, Multi-variable calculus and linear algebra with applications to differential equations

and probability, Wiley, New York, 1969.

[6] S. Boyd and L. Vandenberghe, Convex optimization, Cambridge University Press, 2004.[7] C. B. Boyer, The history of the calculus and its conceptual development, Dover, New York, 1949.

[8] C. B. Boyer, A history of mathematics, Wiley, 1968.

[9] G. Buttazzo, M. Giaquinta and S. Hildebrant, One dimensional variational problems, an introduction, OxfordPress, Oxford, 1998.

[10] H. Brezis, Analyse Fonctionnelle, theorie et applications, Masson, Paris, 1983.

[11] Ph. G. Ciarlet, Mathematical elasticity, Vol 1., three dimensional elasticity, North-Holland, 1988.[12] E. A Coddington, An introduction to ordinary differential equations, Dover, 1961.

[13] E. A Coddington and N. Levison, Theory of ordinary differential equations, McGraw-Hill, 1963.

[14] B. Dacorogna, Introduction to the Calculus of Variations, Imperial College Press, 2004.[15] B. Dacorogna, Direct Methods on the Calculus of Variations, Applied Mathematical Sciences, 78, Springer-Verlag,

Berlin, 1989.

[16] J. Dieudonne, Infinitesimal calculus, Hermann, 1971.[17] L. Evans, Partial differential equations, Amer. Math. Soc., Providence, 1998.

[18] L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, CRC Press, 1992.[19] R. Feynman, R. Leighton and M. Sands, The Feynman Lectures on Physics, Vol II, Addison-Wesly, 1966

[20] P. Freguglia and M. Giaquinta, The early period of the Calculus of Variations, Birkhauser, 2016.

[21] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp-Spaces, Springer Monographs inMathematics. New York, NY: Springer, 2007.

[22] G. Folland, Real Analysis: Modern techniques and their applications, Wiley-Interscience, 1984.

[23] I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood, 1963.[24] M. Giaquinta and S. Hildebrant, Calculus of variations I, II, Grundlehren math. Wiss. 310, 311, Springer, Berlin,

1995.

[25] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003.[26] H. H. Golstine, A history of the calculus of variations from the 17th to the 19th century, Springer, Berlin, 1980.

[27] M. E. Gurtin, An introduction to continuum mechanics, academic Press, New York, 1981.

[28] M. Kot, A first course in the calculus of variations, Student mathematical library, Vol. 72, AMS, Rhode Island,2014.

[29] E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, New York, 1978.[30] G. Leoni, A First Course in Sobolev Spaces (See also: Lecture Notes by this author)

[31] M.I.T Open Course, Convex Analysis

[32] F. Rindler, Introduction to the Modern Calculus of Variations (Lecture Notes)[33] W. Rudin, Functional analysis, McGraw-Hill, New York, 1073.

[34] H. L. Royden, Real analysis, Macmillan Publishing Company, New York, 1988.

[35] R. T. Rockafellar, Convex analysis, Princeton University Press, 1988.[36] H. Sagan, Introduction to the Calculus of Variations, Dover, New-York, 1969.

lecture notes on calculus of variations and partial ... · differential equations (2s-16/17)....

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