Lecture1-3

Intoduction to calculus of variation and

asymptotic analysis: Applications to Solid

Mechanics

O. Pantz

November 22, 2013

2

Contents

1 Introduction 51.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.1 Energy functionnals . . . . . . . . . . . . . . . . . . . . . 61.1.2 Energy depending on a small parameter . . . . . . . . . . 6

2 Existence of minimizers 92.1 Finite case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 String of masses . . . . . . . . . . . . . . . . . . . . . . . 112.2 Infinite case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Existence in Hilbert Spaces . . . . . . . . . . . . . . . . . 132.2.2 Existence in Uniformely Convex Spaces . . . . . . . . . . 132.2.3 Existence in Reflexive Spaces . . . . . . . . . . . . . . . . 132.2.4 Weak topology . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Lp and Sobolev spaces 153.1 Notions of integration . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Elementary properties of Lp spaces . . . . . . . . . . . . . . . . . 173.3 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Application to some Minimization problems . . . . . . . . . . . . 19

4 Optimality conditions and Numerical Methods 214.1 Euler-Lagrange equations . . . . . . . . . . . . . . . . . . . . . . 214.2 Gradient based Algorithm . . . . . . . . . . . . . . . . . . . . . . 21

5 Formal asymptotic analysis 235.1 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 Non linear elastic rod . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2.1 Setting of the problem . . . . . . . . . . . . . . . . . . . . 285.2.2 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Γ-convergence 316.1 Definition and basic properties . . . . . . . . . . . . . . . . . . . 316.2 Lower semicontiuous envelope . . . . . . . . . . . . . . . . . . . . 35

6.2.1 Relaxation in the one dimensional case . . . . . . . . . . . 36

3

4 CONTENTS

6.2.2 Relaxation in the scalar case . . . . . . . . . . . . . . . . 396.2.3 Relaxation in the vectorial case . . . . . . . . . . . . . . . 39

6.3 Application to elastic rods . . . . . . . . . . . . . . . . . . . . . . 406.4 Application to elastic membranes . . . . . . . . . . . . . . . . . . 43

7 Other examples (formal approach) 457.1 Elastic beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.2 Elastic plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.3 Modica Mortola . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Chapter 1

Introduction

The aim of this course is to give basic tools in calculus of variation and asymp-totic analysis. The stress will be put on applications to mechanics, in particularsolid mechanics. Many mechanical problems can be viewed as solution of a min-imization problem. A basic example is given by linear and non linear elasticity,where equilibrium states can be defined as minimizer (possibly local) of theelastic energy of the body. Many other mechanical systems can be described insuch a way: Temperature distribution in a body, viscous fluids, lattices, brittlematerials, mutliphase fluids or martensitic materials to name a few. An impor-tant feature of most of those systems is that they have an infinite number ofdegree of freedom. For instance, in a quasi-static setting, the state of a nonlinearelastic body is given by a map from a reference configuration into the euclidianspace. Contrarly to the finite case, the existence is far from trivial. Such aquestion could appear theoretical. Indeed, the existence of a solution seems tobe justified by the very fact that the model is based on a physical reality thatcould not been denied. Firstly, such attitude is based on the assumption thatthe model is a perfect representation of the reality. This is NEVER the case. Atmost it can be a correct approximation of the reality. Considering the problemof existence from the mathematical point of view offer several insights on thephysical reallity. First, existence or non existence can help to distinghish be-tween reasonable and faulted modelings. Secondly, as we will see, non existencecan result from the overlook of small scale phenomena. Studying minimiza-tion sequences of such non well posed problem can help to identify them andthus either include them into the modeling by adding a term ot the energy orneglecting them by applying a relaxation on the initial energy functional.

Another question is to determine meaningful ways to compute numericallythe minimizers of such functionnals. A most commun approach is based on thewell known gradient method. In a landscape, following the steepest slope willlead you (under certain conditions) to the least height point of the landscape(but in most cases to a local minimum). Anyway, you will only stop once you’vereached a flat area where the step is null. In the context of the minimization ofthe energy of a system, this steepest slope is determined thanks to the differential

5

6 CHAPTER 1. INTRODUCTION

of the energy. Thus minizers are (at least partially) caracterized as roots ofthe differential, which are given by the Euler-Lagrange equation. The Euler-Lagrange equation is

1.1 Examples

1.1.1 Energy functionnals

In this section, Ω will be an open bounded subset of RN and ΓN , ΓD will designsubset of the boundary ∂Ω of the domain, such that ∂Ω = ΓD ∪ ΓN .

Distribution of tempearture We assume that Ω is the domain occupiedby a conductive body of conductivity k. The distribution temperature u isimposed to be equal to uD on ΓD and a thermic flux g is imposed on ΓN with∂Ω = ΓD ∪ ΓN . Then u of temperature is solution of

infv=uD on ΓD

1

2

∫Ω

k|∇v|2 dx−∫

ΓN

gv dx

Non linear elasticity

infψ

1

2

∫Ω

W (∇ψ) dx−∫

ΓN

g · ψ dx

Linear elasticity

infv

1

2

∫Ω

2µ|e(v)|2 + λ(div(v))2 dx−∫

ΓN

g · v dx

Stokes equation

infv

∇·v=0

1

2

∫Ω

ν|∇v|2 dx−∫

Ω

f · v dx

1.1.2 Energy depending on a small parameter

An example: String of atoms

infψ∈RN×d

1

2

∑i

W

(|ψi+1 − ψi|

ε

)ε−

∑i

(fi · ψi)ε

Thin plate

1

2

∫Ωε

Wε(∇ϕ) dx+

∫Ω

Fε(ϕ) dx+

∫Γ++Γ−

G(ϕ) ds.

with Fε(s) = ε−1g or Fε(s) = ε−1(As.s), with g ∈ RN (dead load) and Apositive matrix. When the thickness ε goes to zero, different models are derived

1.1. EXAMPLES 7

depending on the scale order of the stored energy Wε with with respect tothe thickness. Let W 0 be frame indifferent, W 0 ≥ 0, W 0(F ) = 0 iff F is arotation and W 1 be frame indifferent, W 1 ≥ 0, W 1(F ) = 0 iff det(F ) = 1 andF1 ∧ F2 = F3.

• Membrane model Wε(F ) = ε−1W 0(F )

• Isometric beding model Wε(F ) = ε−3W 0(F )

• Helfrich model (for bilogical vesicles) Wε(F ) = ε−3W 1(F )

• Red Blood Cell model Wε(F ) = ε−1W 0(F ) + ε−3W 1(F )

• Von Karaman model Wε(F ) = ε−5W 0(F )

Homogenization Example in linear elasticity. Let µ and λ be Y-periodicmaps with positive values, Y being the unit square of RN , meaning that µ(x+ei) = µ(x) for all x ∈ RN and for all i ∈ 1, · · · , N, (ei) being the classicaleuclidian basis of RN (same thing for λ).

infv

1

2

∫Ω

2µε|e(v)|2 + λε(div(v))2 dx−∫

ΓN

g · v dx

with µε(x) = µ(x/ε) and λ(x) = λ(x/ε).

Modica-Mortola

infv

ε

2

∫Ω

|∇v|2 dx+ ε−1

∫Ω

(|v|2 − 1)2 dx.

8 CHAPTER 1. INTRODUCTION

Chapter 2

Existence of minimizers

In this section, we interest ourselves to the existence (or non existence) of solu-tion of problems of the form u ∈ K

J(u) = infv∈K

J(v),

where K is the subset of a vectorial space. We first begin with the finite dimen-sional case (that is K ⊂ RN ), which is quite trivial, but enable us to introducesome basic and useful notions used in the following: infinity at infinity, semicontinuity, minimization sequences. We also provide a simple physical example(string of masses) that will be used in the chapter on asymptotic anaylsis.

2.1 Finite case

Definition (Infinite at infinity) A map J : K → R, with K ⊂ E vectorialspace, is said to be infinite at infinity if and only if for any sequence un ofelements of K such that ‖un‖E

n→∞−−−−→∞, then

limn→∞

J(un) =∞.

Note that if K is bounded, then J is always infinite at infinity.

Definition (Lower semi continuity) A map J from K ⊂ E topologial spaceinto R is said to be lower semi continous is and only if, for every λ

Eλ(J) = J−1((−∞, λ])

is a closed subset of E.

Exercise 1. Prove that the supremum of lower semicontinuous functions islower semi continious.

9

10 CHAPTER 2. EXISTENCE OF MINIMIZERS

Definition (Lim inf) Let xn be a sequence in R, we set

lim infn→∞

xn = limn→∞

infxp such that p > n.

Contrarly to the limit, the lim inf of a sequence always exists (even if possiblyequal to ±∞).

Exercise 2. Prove that the lim inf always exists.

Proposition 2.1.1. Let un be a sequence of elements of a topological space Econverging toward an element u ∈ E and J a semicontinuous functional, then

J(u) ≤ lim infn

J(un). (2.1)

Proof. Letλ = lim inf

nJ(un).

If λ = +∞, there is nothing to prove. Otherwise, for all ε > 0, there existsN ∈ N such that for all n > N ,

infJ(up) such that p > n ≤ λ+ ε.

That is, for all p,J(up) ≤ λ+ ε,

that is up ∈ Eλ+ε(J). The functional J being lower semicontinuous, the setEλ+ε(J) is closed. Thus, letting p goes to infinity, we get u ∈ Eλ+ε(J). In otherwords,

J(u) ≤ λ+ ε

for all ε > 0. It follows that J(u) ≤ ε.

Exercise 3. Let E be a metric space and J be a map from E into R. Provethat J is lower semicontinuous iif for all u ∈ E and for all converging sequenceun toward u,

J(u) ≤ lim inf J(un),

Proposition 2.1.2. Let K be a non empty closed subset of RN and J : K → Ra lower semicontinuous map, infinite at infinity. Then there exists u ∈ K suchthat

J(u) = infv∈K

J(v).

Proof. Let un be a minimization sequence of J on K (such a sequence doesalways exists, see Exercise 4). That is (un) ∈ KN and J(un) → infK J . AsJ is infinite at infinity and J(un) bounded from above, we deduce that un isbounded by a constant R. Thus un belongs to the compact set K ∩BR, whereBR is the ball centered at the origin of radius R. It follows that there existsϕ : N→ N strictly increasing, and u ∈ K∩BR such that uϕ(n) converges towardu when n goes to infinity. Finally, from the lower semi continuity of J

J(u) ≤ lim inf J(uϕ(n)) = lim J(un) = infKJ.

As u ∈ K, it is a minimizer of J on K.

2.1. FINITE CASE 11

Exercise 4. Let F be a non empty subset of R, bounded from below. Prove thatit admits a (unique) greater lower bound. Deduce that if J is a map from a setE into R, it always admits a minimizing sequence, or in other words that thereexists a sequence un ∈ E such that J(un) converges toward the infimum of J onE.

2.1.1 String of masses

We consider a string of N atoms moving in Rd. We set ε = 1/(N + 1), which isof the same scale order than the interactomic distance. The energy of the wholesystem is the sum between an elastic energy due to the interaction betweennearest neighbor atoms and an external potential. The elastic part is discribedby a stored energy W : Rd → R+ whereas the external part is assumed to derivefrom external dead body loads f ∈ Vε, where Vε = (Rd)N is the set of admissiblestates of the system. We only consider here the static case, where the state ofthe system is given by a deformation u ∈ Vε, where ui (with i = 1, · · · , N),stands for the position of the i-th atom of the string in Rd. For all deformationu ∈ Vε of the string of atoms, we introduce the energy

Jε(u) =1

2

N∑i=0

W

(ui+1 − ui

ε

)ε−

N∑i=1

fi · uiε, (2.2)

with the convention u0 = u− and uN+1 = u+, where u− and u+ are the imposedpositions at the ends of the string.

The internal energy of the string is assume to be frame indifferent. Meaningthat the elastic energy of the strings is invariant under rotation.

Definition (Frame indifference) A stored energy W from Rd×m into R issaid to be frame indifferent if for any rotation R ∈ SO(d), we have

W (RF ) = W (F ) for all F ∈ Rd×m.

Without any other asumptions on W , the energy is not necessarily boundedfrom below.

Exercise 5. Prove in the case N = 1, that Jε is bounded from below for allf ∈ Rd iif W (F ) is growing faster than any linear function of the norm of F ,or in other words, that for all β > 0, there exists C(β) ∈ R such that for allF ∈ Rd,

W (F ) ≥ C(β) + β|F |.

Proposition 2.1.3. Let Jε defined by (2.2), with W being lower semicontinu-ous, frame indifferent and growing faster than any linear function of the norm.Then it reaches its minimum on Vε.

Proof. We can now tackle the general case. Let us try to prove the existence ofsolution for any given N . Let us consider once again a minimization sequenceun


M ≥ 1

2

∑i

W

(uni+1 − uni

ε

)ε−

∑i

fi · uiε

≥ NC(β)/ε+ β∑i

|uni+1 − uni |ε

−max(|fi|)∑i

|uni |

Moreover, for any u, we have for all i,

|ui| ≤∑i

|ui+1 − ui|. (2.3)

Thus, we get

M ≥ 1

2

∑i

W

(uni+1 − uni

ε

)ε−

∑i

fi · uiε

≥ NC(β)/ε+ β∑i

|uni+1 − uni |ε

−max(|fi|)N∑i

|uni+1 − uni |

and

M ≥ 1

2

∑i

W

(uni+1 − uni

ε

)ε−

∑i

fi · uiε

≥ NC(β)/ε+ (β −Nεmax(|fi|))∑i

|uni+1 − uni |ε

Chosing β big enough, we get that

‖un‖1 =∑i

|uni+1 − uni |ε

is bounded, and from (??) that un is bounded. Finally existence do follow fromthe l.s.c. assumption made on W .

Example We can chose for instance W (s) = ||s|2 − 1|2, that is

Jε(u) =1

2

∑i

(∣∣∣∣ui+1 − uiε

∣∣∣∣2 − 1

)2

ε

2.2 Infinite case

In this section, we are going to give few existence results in Banach spaces.Roughly speaking, the existence of a minimizer of a functional J : E → R isensured if J is continuous, infinite at infinity and CONVEXE (provided thatE is a nice enough Banach space). We are going to study three different cases,depending on the properties of the space E, from the more particular to themore general. More precisely, we will first focus on the case where E is aHilbert space, then

2.2. INFINITE CASE 13

2.2.1 Existence in Hilbert Spaces

2.2.2 Existence in Uniformely Convex Spaces

2.2.3 Existence in Reflexive Spaces

2.2.4 Weak topology

Let us recall some definition about weak conergence in Banach spaces.

Definition (Weak convergence) Let E be a Banach space and un a sequencein E. We say that un convergences weakly toward u ∈ E in E

un u,

if and only ifL(un)→ L(u),

for all L ∈ E′.

Definition (Weak-* convergence) Let E be a Banach space and Ln a se-quence in E′. We say that Ln convergences weakly-* toward L ∈ E′ in E′

Ln ∗L,

if and only ifLn(u)→ L(u),

for all u ∈ E.

We recall that the norm on E′ is defined by

‖L‖E′ = supx∈E ‖x‖E=1

L(x).

Theorem 2.2.1 (Banach-Alaoglu-Bourbaki). The unit ball of E′ is compactfor the weak-* topology.

Proof. We propose to perform the proof by assuming that the space E is separa-ble. In the general case, the proof can be found in Brezis [3]. As E is separable,there exists a sequence of elements xn dense in E. Now, let us consider a se-quence Ln in the unit ball of E′. We construct a sequence of strictly growingmaps from N into N (ϕn) such that for all n, Lψn(k)(xp) is converging for allp ≤ n, where ψn = ϕ0 · · · ϕn. As Ln(x0) is bounded, it is clear that wecan extract a subsequence Lϕ0(k) such that Lϕ0(k) is convergent. Next for alln > 0, assuming that (ϕp)p<n have been build, then Lψn−1(k)(xp) is convergingfor all p < n and Lψn−1(k)(xn) is bounded. Thus, we can extract a subsequenceLψn−1ϕn(k) such that Lψn−1ϕn(k)(xn) is convergent. Obviously, we still haveLψn−1ϕn(k)(xp) is still convergent for all p < n. Finally, we set ϕ(n) = ψn(n).The sequence ϕ(n) is strictly growing from N into N and Lϕ(n)(xp) is convergingfor all p ∈ N. Finally, using the fact that Lϕ(n) is uniformely bounded and that


(xp) is dense in E, we can prove that Lϕ(n)(x) is a Cauchy sequence (and thusconvergent) for all x ∈ E. Let us denote L the limit. It only remains to provethat L is linear and continuous (what is easy). Moreover, we obtain that

‖L‖E′ ≤ lim inf ‖Ln‖E′ .

Proposition 2.2.2. If E is a Banach space, then for all x ∈ E,

‖x‖E = supL∈E′ ‖L‖E′=1

L(x).

Proof. We have L(x) ≤ ‖L‖E′‖x‖E . It follows that for all continuous linearform L such that ‖L‖E′ = 1,

‖x‖E ≥ L(x),

thus‖x‖E ≥ sup

L∈E′ ‖L‖E′=1

L(x).

The converse inequality is far less trivial and is based on the Hahn-BanachTheorem (see Brezis [3]) for more details.

Exercise 6. Prove the Proposition 2.2.2 when E is a Hilbert Space.

Proposition 2.2.3. Let un be a sequence wealky converging toward u, then

‖u‖E ≤ lim inf ‖un‖E .

Proof. We have‖u‖E = sup

L∈E′ ‖L‖E′=1

L(u).

Thus, u is the supremum of continuous (and thus lower semicontinuous) func-tions for the weak topology. In consequences, it is iteself lower semicontinuousfor the weak topology.

Note that the same result holds for the weak-* convergence. Moreover, wecan extend this result to a more general case.

Proposition 2.2.4. Let E be a Banach space and J a convex semicontinu-ous function (for the strong topology), then J is semicontinuous for the weaktopology.

This result is based on the Hahn-Banach Theorem (see once again Brezis[3]) for more details.

Using this result, and the Banach-Alaoglu-Bourbaki Theorem, it is easy toprove the existence Theorem of minimizers .

Chapter 3

Lp and Sobolev spaces

Let us consider the minimization problem of

J(u) =1

2

∫Ω

|∇u|2 + |u|2 dx−∫

Ω

fu dx.

For the moment, we can not apply any of the Theorems of the previous section.The main reason is that we do not have the correct Banach space to apply oneof the previous result. Indeed, to define the function J we have to assume thatu is derivable and thus to search for a minimizer over C1(Ω). Unfortunately,J is not infinite at infinity for such a choice. In this particular case, we cannevertheless use the following small trick. Assume that u is derivable, then

1

2

∫Ω

|τ |2 + |u|2 dx−∫

Ω

fu dx+

∫Ω

(τ −∇u) · σ

We minimize with respect to τ and get τ = σ, leading to a min max problem of

1

2

∫Ω

−|σ|2 + |u|2 dx−∫

Ω

fu dx+

∫Ω

−∇u · σ

=1

2

∫Ω

−|σ|2 + |u|2 dx−∫

Ω

fu dx+

∫Ω

u(∇ · σ) dx.

Thus, we have

J(u) = supσ∈C∞0 (Ω)

1

2

∫Ω

−|σ|2 + |u|2 dx−∫

Ω

fu dx+

∫Ω

u(∇ · σ) dx

The functional on the left and side are all convex and lower semicontinuousin L2(Ω). Thus, it follows that J is itself convex and semicontinuous in L2

(possibly taking the value +∞). Moreovoer, it is clearly infinite at infinity andJ admits a minimizers on L2. Nevertheless, this approach raised to problems.The first one is that it is not obvious to recover the Euler-Lagrange equations

15

16 CHAPTER 3. LP AND SOBOLEV SPACES

satisfied by u. Another approach consists to endowe the space of the continuousfunctions with the norm

‖u‖2H1(Ω) =

∫Ω

|∇u|2 + |u|2 dx.

With such a space, J is clearly continuous, convex and infinite at infinity. It isalso clearly a Hilbert space, included in L2(Ω). So once again, we obtain theexistence of solutions. The Sobolev space H1(Ω) is precisely the obtained space.A natural question consists in trying to identify such a space. In fact, we have

H1(Ω) = u ∈ L2(Ω) such that there exists τ ∈ L2(Ω)N such that

for all σ ∈ C∞0 (Ω) we have

∫Ω

u∇ · σ dx = −∫

Ω

τ · σ dx.

It is easy to check that the closure of the C1(Ω) functions is included in thisspace

3.1 Notions of integration

Let p be an integer, p ≥ 1. We recall that Lp(Ω) is the space of mesurablefunctions f such that ∫

Ω

|f(x)|p dx <∞,

endowed with the norm

‖f‖Lp(Ω) =

(∫Ω

|f(x)|p dx)p

.

Theorem 3.1.1 (Beppo-Levi monotone convergence Theorem). Let (fn) be agrowing sequence in L1(Ω) such that

supn

∫Ω

fn dx <∞,

then fn(x) convergences for almost all x ∈ Ω toward a finite limit f(x). More-over, f ∈ L1(Ω) and fn convergences toward f in L1(Ω).

Lemma 3.1.2 (Fatou). Let (fn) be a sequence of non negative functions inL1(Ω), we have ∫

Ω

lim infn

fn dx ≤ lim inf

∫Ω

fn dx.

An almost straighforward consequence of the Fatou’s Lemma is the widelyused Lebesgue dominated convergence Therorem

Theorem 3.1.3 (Lebesgue dominated convergence Theorem). Let (fn) be asequence in L1(Ω) such that

3.2. ELEMENTARY PROPERTIES OF LP SPACES 17

• fn(x)→ f(x) for almost all x ∈ Ω.

• There exists g ∈ L1(Ω) such that |fn(x)| ≤ g(x) almost everywhere (a.e.).

Then f ∈ L1(Ω) and fn convergences toward f in L1(Ω).

Note that this Theorem remains true when replacing L1 by Lp.

Theorem 3.1.4 (Converse Lebesgue Theorem). Let p ∈ [1,∞) and fn be asequence of functions in Lp(Ω) congerging toward an element f in Lp(Ω), thenwe can extract a subsequence fn′ such that

• There exists g ∈ Lp(Ω) such that |fn′(x)| ≤ g(x) a.e.

• fn′(x)→ f(x) a.e.

3.2 Elementary properties of Lp spaces

We already gave the definition of the spaces Lp(Ω) for 1 ≤<∞. For p =∞, wedefine the space L∞(Ω) as the set of mesurable functions such that

‖f‖L∞(Ω) = infCsuch that |f(x)| ≤ C for almost all x ∈ Ω <∞.

Theorem 3.2.1 (Holder ineqalities). Let f ∈ Lp(Ω) and g ∈ Lp′(Ω) with

p ∈ [1,∞] and 1/p+ 1/p′ = 1, then∫Ω

|fg| dx ≤ ‖f‖Lp(Ω)‖g‖Lp′ (Ω)

Theorem 3.2.2 (Fischer-Riesz). Lp(Ω) is a Banach space for all 1 ≤ p ≤ ∞.

Theorem 3.2.3. Lp(Ω) is uniformly convex thus reflexive for all 1 < p <∞.

Theorem 3.2.4 (Riesz representation Theorem). Let 1 ≤ p < ∞, then for allL ∈ Lp(Ω)′, there exists an unique u ∈ Lp′(Ω), with 1/p+ 1/p′ = 1 such that∫

Ω

uv dx = L(v) for all v ∈ L(Ω).

Theorem 3.2.5 (density). The space C∞0 (Ω) is dense in Lp(Ω) for all 1 ≤ p <∞.

Theorem 3.2.6 (separability). Lp(Ω) is separable for all 1 ≤ p <∞.

3.3 Sobolev Spaces

Definition (Definition) Let 1 ≤ p ≤ ∞. The Sobolev space W 1,p(Ω) isdefined by

W 1,p(Ω) = u ∈ Lp(Ω) such that there exists σ ∈ L(Ω)N such that for all τ ∈ C∞0 (Ω), we have

∫Ω

u∇·τ dx = −∫

Ω

∇u·τ dx


We denote ∇u = τ . Moreover, we set

‖u‖W 1,p(Ω) = ‖u‖Lp(Ω) + ‖∇u‖Lp(Ω)N .

In the case p = 2, we usually use the notation H1(Ω) instead of W 1,p(Ω).

Proposition 3.3.1. The space W 1,p(Ω) is a Banach space. For 1 ≤ p <∞, itis separable, for 1 < p <∞ it is reflexive. Moreover, H1(Ω) is a Hilbert space.

Proposition 3.3.2 (Density). If Ω is a Lipschitzian open set, then C∞0 (Ω) isdense in Lp(Ω) for all 1 ≤ p <∞.

Proposition 3.3.3 (Sobolev injection). Let Ω = RN or Ω be a regular opensubset of RN , with a bounded boundary, then

• If 1/p∗ = 1/p−1/N > 0, then W 1,p(Ω) is continuously included in Lp∗(Ω).

• If 1/p− 1/N = 0, then W 1,p(Ω) is continuously included in Lq(Ω) for allp ≥ q <∞.

• If 1/p− 1/N < 0, then W 1,p(Ω) is continuously included in L∞(Ω).

Proposition 3.3.4 (Rellich-Kondrachov). Let Ω ∈ RN be a regular open boundedsubset of Rn, then

• If 1/p∗ = 1/p−1/N > 0, then W 1,p(Ω) is continuously, compactly includedin Lq(Ω) for all 1 ≤ q < p∗.

• If 1/p − 1/N = 0, then W 1,p(Ω) is continuously, compactly included inLq(Ω) for all 1 ≤ q <∞.

• If 1/p − 1/N < 0, then W 1,p(Ω) is continuously, compactly included inL∞(Ω).

Proposition 3.3.5 (Trace). Let Ω be a Lipschitzian open set and p ∈ [1,∞),then the map from W 1,p(Ω) into Lp(∂Ω) is continuous.

Proposition 3.3.6 (Poincare inequality). Let Ω be an open set of RN , boundedin one direction then there exists C such that

∀u ∈W 1,p0 (Ω)‖u‖Lp(Ω) ≤ C‖∇u‖Lp(Ω)N

There exists other useful versions of the Poincare inequality

Proposition 3.3.7 (Poincare inequality). Let Ω be a bounded connected openset of RN , and ΓD be a subset of ∂Ω of non zero N − 1 measure. Then thereexists C such that for all u ∈W 1,p(Ω), such that u = 0 on ΓD, we have

‖u‖Lp(Ω) ≤ C‖∇u‖Lp(Ω)N

Another usefull version

Proposition 3.3.8 (Poincare-Wirtinger inequality). Let Ω be a bounded con-nected open set of RN . Then there exists C such that for all u ∈W 1,p(Ω), suchthat u = 0 on ΓD, we have

‖u−m‖Lp(Ω) ≤ C‖∇u‖Lp(Ω)N ,

where m is the mean value of u over Ω.

3.4. APPLICATION TO SOME MINIMIZATION PROBLEMS 19

3.4 Application to some Minimization problems

• Let Ω be open subset of RN and

J(u) =1

2

∫Ω

|∇u|2 + u2 dx−∫

Ω

fu dx

The functional J is convexe, infinite at infinite, continious over H1(Ω),Hilbert space and thus admits a minimizer. Moreover, as J is strictlyconvexe, it it unique.

• Let Ω be a bounded regular open subset of RN and

J(u) =1

2

∫Ω

|∇u|2 dx−∫

Ω

fu dx et u = 0 sur ∂Ω

The functional J is convexe, continuous over H10 (Ω) , Hilbert space, and

infinite at infinity, thanks to the Poincare inequality. Thus, it admits aunique minimizer. Moreover, the minimizer is unique (as the problem islinear, it is enough to check that for f = 0, u = 0 is the unique solution).

• Let Ω be a bounded subset of RN , F convexe continuous such that

J(u) =1

2

∫Ω

|∇u|2 + u2 dx+

∫Ω

F (u) dx

First, a F is convex,F (s) ≥ F ′(0)s+ F (0).

It follows that∫Ω

F (u) dx ≥ −|F ′(0)|‖u‖L1(Ω) + |Ω|F (0) ≥ C1‖u‖L2(Ω) + C2.

Thus J(u) goes to infinity as ‖u‖H1(Ω) goes to infinity. Next, let us assumethat

F (u) ≤ C(|u|2 + 1),

then, using the Lebesgue Theorem (and the partial converse Theorem),we have that

u 7→∫

Ω

F (u) dx

is continuous for the L2(Ω) norm. Finally, J is strictly convex, continuous,infinite at infinity and thus admits a unique minimizer.

It can be check that the convexity assumption on F can be dropped, bykeeping only the assumptions

C1s+ C0 ≤ F (s) ≤ C(|s|2 + 1).

We still have that J is infinite at infinity, so that the minimization se-qences are compact for the weak topology of H1(Ω). Let un be a sequence


converging for the weak topology toward u. From the Rellich Theroem, upto the extraction of another subsequence, it converges strongly in L2(Ω),so that F (un) converges strongly toward F (u). Using the lowar semi con-tinuity of the norm, we conclude that u is a minimizer.

• Let us consider the functionnal

J(u) =1

2

∫Ω

F (u) dx

and asssume that|F (s)| < C(1 + |s|p),

with p ∈ (1,∞). If F is convex, continuous, then J is infinite at infinity,convex and continuous. Thus, it admits a minimizer of Lp(Ω). Conversely,if J est lower semicontinuous for the weak topology, then it is convex.Indeed, let a and b in R and let us define the one periodic function on Rby

T (s) = a if s ∈ (0, θ)b if s ∈ (θ, 1)

then un(x) = T (nx1) weakly converges toward the constant map θa+(1−θ)b and assuming that J is lower semicontinuous for the weka topology, itfollows that

F (θa+ (1− θ)b) ≤ θF (a) + (1− θ)F (b).

(il faut faire un truc sur l’enveloppe semi infrieure).

•J(u) =

1

2

∫Ω

W (∇u) dx

Dans W 1,p(Ω), u = 0 sur le bord et W convexe/non convexe ?

•J(u) =

1

2

∫Ω

W (∇u) dx+

∫Ω

F (u) dx

W convexe et F non convexe.

• Stokes

• Elasticit Linaire (ingalit de Korn)

Chapter 4

Optimality conditions andNumerical Methods

4.1 Euler-Lagrange equations

4.2 Gradient based Algorithm

21

22CHAPTER 4. OPTIMALITY CONDITIONS ANDNUMERICALMETHODS

Chapter 5

Formal asymptotic analysis

5.1 Homogenization

We consider a conductive body Ωε ∈ RN . We suppose that Ωε is made oftiny holes periodically displayed inside a domain Ω of RN . More precisely, weassume that there exists an Y -periodical connected open subset ω of RN , whereY = (0, 1)N such that

Ωε = Ω ∩ εω.

We denote by uε the potential at equilibrium inside Ωε, assuming that we applya volumic flux f ∈ L2(Ω) with Dirichlet boundary conditions of ΓεD = ∂Ω∩Ωε.In other words, uε is the minimizer of

Jε(vε) =1

2

∫Ωε

A∇vε · ∇vε dx−∫

Ωε

fvε dx,

over the set

Vε = vε ∈ H1(Ω− ε) : vε = 0 on ΓεD.

We seek for the limit of uε as ε goes to zero. To this end, we are going to assumethat uε admits an asymptotic expansion.

Ansatz 5.1.1. There exists a sequence u = (ui)i∈N ∈ V , with

V :=v ∈ H(Ω× ω)N such that for all i,

vi(x, y) = 0 for all (x, y) ∈ ∂Ω× ωvi(x, ·) is Y periodical with respect to the second variable

such that

uε(x) =

∞∑i=0

εiui(x, x/ε).

23

24 CHAPTER 5. FORMAL ASYMPTOTIC ANALYSIS

For all sequence v ∈ V , we introduce the energy

Jε(v) = Jε(vε),

where

vε(x) =

∞∑i=0

εivi(x, x/ε).

We now can rewrite the energy Jε

as an asymptotic exanpsion in the follow-ing form,

Jε(v) =

∑k∈Z

Jε

k(v). (5.1)

Now we identifiy the terms of this expansion. We set vi = 0 for all i < 0. Wehave

Jε(v) =

1

2

∫Ωε

A∇vε · vε dx−∫

Ω−εfvε dx

=1

2

∫Ωε

A∇

(∑i

εivi(x, x/ε)

)·

(∑i

εivi(x, x/ε)

)dx−

∫Ωε

f

(∑i

εivi(x, x/ε)

)dx

=1

2

∑i,j

εi+j∫

Ωε

A(∇xvi(x, x/ε) + ε−1∇yvi(x, x/ε)) · (∇xvj(x, x/ε) + ε−1∇yvj(x, x/ε)) dx−∑k

εk∫

Ωε

fvk(x, x/ε) dx

=∑k∈Z

∑i, j

i+ j = k

(1

2

∫Ωε

A(∇xvi(x, x/ε) · ∇xvj(x, x/ε) + 2A(∇xvi(x, x/ε) · ∇yvj+1(x, x/ε)A(∇yvi+1(x, x/ε)∇yvj+1(x, x/ε) dx−∫

Ωε

fvk(x, x/ε) dx).

We thus obtain the expansion (5.1) with

Jεk(v) =1

2

∑i+j=k

∫Ωε

A∇xvi(x, x/ε) · ∇xvj(x, x/ε) + 2A∇xvi(x, x/ε) · ∇yvj+1(x, x/ε)A∇yvi+1(x, x/ε)∇yvj+1(x, x/ε) dx−∫

Ωε

fvk(x, x/ε) dx

=1

2

∑i+j=k

∫Ωε

A(∇xvi +∇yvi+1) · (∇xvi +∇yvi+1)(x, x/ε) dx−∫

Ωε

fvk(x, x/ε) dx.

Now, we need a little lemma

Lemma 5.1.2. Let g ∈ C(Ω, L1(ω)) be Y -periodic map with respect to the secondvariable, we have

limε→0

∫Ωε

g(x, x/ε) dx =

∫Ω

∫Y ∩ω

g(x, y) dy dx.

Proof. We start the analysis, assuming that g(x, y) depends only of y. To sim-plify the notation, we will denote g(y) ∈ L1(ω) in place of g(x, y). Moreover,we assume g to be non negative. We have∫

Ωε

g(x/ε) dx = εN∫ε−1Ω∩ω

g(y) dy.

5.1. HOMOGENIZATION 25

We denote by (Cεi )i∈ZN the set of cubes of the form ε((−1, 1)N + i). Let I−ε theset of indices i such that Cεi is included in Ω, and I+

ε the set of indices i suchthat the intersection between Ω and Cεi is not empty. We have

∪i∈I−ε Cεi ∩ ω ⊂ Ωε ⊂ ∪i∈I+ε .

As g is assumed to be nonnegative, we have

εN∑i∈I−ε

∫ε−1Cε

i ∩ωg(y) dy ≤

∫Ωε

g(x/ε) dx ≤ εN∑i∈I−ε

∫ε−1Cε

i ∩ωg(y) dy

and

εN |I−ε |∫Y ∩ω

g(y) dy ≤∫

Ωε

g(x/ε) dx ≤ εN |I+ε |∫Y ∩ω

g(y) dy.

Finally, εN |I±ε | converges toward the volume of Ω. It follows that∫Ωε

g(x/ε) dx→ |Ω|∫Y ∩ω

g(y) dy.

It is clear that this result still stands if g is not of constant sign (we just haveto apply the previous analysis to g+ = max(g, 0) and g− = min(g, 0) and to usethat g = g+ + g−.It remains to consider the case where g depends also on x. First, let us asumethat g is constant by part with respect to x, that there exists a finite familly

Ωi of regular disjoint open subset of Ω such that ∪i∈IΩi

= Ω, such that for allx ∈ Ωi

g(x, y) = gi(y),

with gi ∈ L1(ω) is Y -periodic. Then∫Ωε

g(x, x/ε) dx =∑i∈I

∫Ωi

ε

gi(x/ε) dx→∑i∈I|Ωi|

∫Y ∩ω

gi(y) dy =

∫Ω

∫Y ∩ω

g(x, y) dx dy.

By density, the result can be extend to any g ∈ C0(Ω, L1(ω)) periodic withrespect to the second variable.

We setJk(v) = lim

ε→0Jεk(v).

From the lemma, we deduce that

Jk(v) =∑i+j=k

1

2

∫Ω×Y ∩ω

A(∇xvi+∇yvi+1)·(∇xvj+∇yvj+1), dx dy−∫

Ω×Y ∩ωfvk dx dy.

For all k < 2 we have Jεk = 0. We set

Mk = v ∈Mk−1 such that Jk(v) = infw∈Mk−1

Jk(w),

with M−3 is defined as all the regluar maps from Ω × ω into R, Y -periodicalwith respect to the second variable.


Lemma 5.1.3. Let k we such that for all ε > 0 and all l < k, Jεk(v) is inde-pendant of v ∈Mk+1, then u belongs Mk+1.

Proof. It suffies to prove it recursively. Obviously, the Lemma is true for k = −3.Now, let us assume that the Lemma is true for all l ≤ k. Then u ∈Mk and forall v ∈Mk and for all ε > 0, we have

Jε(u) ≤ Jε(v).

Using the asumptotic expanson of Jε

we get∑l

εlJεl (u) ≤∑l

εlJεl (v).

From the assumption that Jεl (v) is independant of v ∈Mk for all l ≤ k, we get∑l≥k+1

εlJεl (u) ≤∑l≥k+1

εlJεl (v).

Dividing this inequality by εk+1, and letting ε goes to zero, we obtain that

Jk+1(u) ≤ Jk+1(v)

and thus u ∈Mk+1.

We denote by (Pk) the minimization problem of Jk over Mk−1. We arenow in position to determine the limit behavior of uε by recursively solvingthe problems (Pk) for k ≥ −2. At each iteration, we have to check that theassumption of the Lemma 5.1.3 is satisified.

Problem (P−2) We have

J−2(u) = sumi+j=k1

2

∫Ω×Y ∩ω

A∇yv0 · ∇yv0, dx dy.

The minimizer of J−2 is trivial and is achieved on M−2 of asymptotic expansionsv such that v0 is indendant of the second variable. In the following, v0(x, y) willbe simply denoted v0(x).

Problem (P−1) Not that Jε−2(v) = 0 for all v ∈ M−2. Thus the assumptionof the Lemma 5.1.3 are satisified and u belongs to M−1. Next,

J−1(v) =

∫Ω×Y ∩ω

A∇xv0 · ∇yv0, dx dy.

Thus, J−1 is constant equal to zero over M−2 and M−1 = M−2.

5.1. HOMOGENIZATION 27

Probleme (P0) Once again, it is easy to check that the assumption of theLemma 5.1.3 are satisifed for k = −1. Thus, u belongs to M0. Next, for all∈M−1, we have

J0(v) =1

2

∫Ω×Y ∩ω

A(∇xv0 +∇yv1) · (∇xv0 +∇yv1), dx dy −∫

Ω×Y ∩ωfv0 dx dy.

Thus, it depends only on v0(x) and v1(x, y). We set

I0(v0) = infv1

1

2

∫Ω×Y ∩ω

A(∇xv0+∇yv1)·(∇xv0+∇yv1), dx dy−∫

Ω×Y ∩ωfv0 dx dy.

Obviously, infM−1 J0 = infv0∈H10 (Ω) I0(v0). The second integral in the expression

of I0 is independant of v1 while to the infimum of the first integral can beperfomed pointwise with respect to the x variable, that is

I0(v0) = infv1

1

2

∫Ω

(inf

v1 Y -periodic

∫ω∩Y

A(∇xv0(x) +∇yv1(y)) · (∇xv0(x) +∇yv1(y)), dy

)dx−

∫Ω

f∗v0 dx,

where

f∗(x) := |ω ∩ Y |f(x)

Let us introduce the symetric matrix A∗ defined by

A∗ξ · ξ := infv1 Y -periodic

∫ω∩Y

A(ξ +∇yv1(y)) · (ξ +∇yv1(y)), dy (5.2)

Using the fact that ω is simply connected, we can prove that the infimum isreached for a unique v1. Moreover, the solution is linear with respect to ξ sothat A∗ξ · ξ is indeed bilenar with respect to ξ, so that there exists indeed aunique symetric matrix A∗ defined by (??). More precisely, we have for allY -periodic test function w1,∫

ω∩YA(ξ +∇yv1(y)) · ∇yw1(y), dy = 0.

Thus, it suffies to solve the minimization problem for all ξ in the euclidan base(e1, · · · , en) to compute the full matrix A∗. Let vi be the solution for ξ = ei.We have

A∗i,j =

∫ω∩Y

A(ei +∇yvi(y)) · (ej +∇yvj(y)), dy.

Finally, we obtain that u0 is the minimizer over H10 (Ω) of

I0(v0) = infv1

1

2

∫Ω

A∗∇xv0 · ∇xv0 dx−∫

Ω

f∗v0 dx.


5.2 Non linear elastic rod

In this section, we derive a one dimensional modeling for a nonlinear elasticrod from nonlinear three dimensional elasticity. Once again, we are going toadopt a formal approach, assuming that the sequence of minimizers ϕε admitsan asymptotic expansion. Unfortunatly, the obtained limit is not the correctone, because the limit problem is not well posed. The correct limit can beobtained by a mere relaxation of the formal obtained limit. In other words, thecorrect limit is simply the lower semicontinuous enveloppe of the formal limit.This will be more rigourously proved in the next section using the notion ofΓ-convergence. Note that it is not necessary important to understand in fulldetails the rigourous derivation of the one dimensional modeling presented inthe next section. But the reader must fully understand why both formal andrigourous limit energy do differ.

5.2.1 Setting of the problem

We consider an elastic rod of section εω of length L, where ω is a regularbounded open subset of R2, ε is a small parameter. The natural configurationof the body is defined by

Ωε = εω × (−L,L).

Moreover, we assume that the rod is dubmitted to volumic body loads fε andthat it is made of a hyperelastic material of stored energy Wε : R3 ×R3 7→ R+.Finally, we apply Dirichlet boundary conditions on the part ΓDε = εω×−L∪L of the body. The deformation at equilibrium is thus the minimizer of

Jε(ψε) =

1

2

∫Ωε

Wε(∇ψε) dx−∫

Ωε

fε · ψε dx

over the deformations ψε such that

ψε(x) = φε(x) for all x ∈ ΓDε ,

where φε are given Dirichlet conditions. For sake of simplisity, we assume thatonly rigid motions are applied at the ends of the rods, that is that there exsitsa rotations R± and vectors c± ∈ R3 such that

φε(x) = R± x+ c± for all x ∈ ω × ±L.

It remains to precise how Wε and fε do scale as ε goes to zero. We assume thatthey both scale in ε−2, and that there exists W and f such that

Wε(F ) = ε−2W (F ) for all matrix F ∈ R3 × R3

andfε(x) = ε−2f(x) for all x ∈ Ωε.

We seek for the limit behavior of the minimizer ϕε of the energy as ε goesto zero. To this end, we are going to assume, as in the case of homogenization,that the minimizers do admit a particular asymptotic expansion.

5.2. NON LINEAR ELASTIC ROD 29

Ansatz 5.2.1. There exists ϕ = (ϕi)i∈N, sequence of ”regular” maps fromΩ = ω × (0, L) into R3 such that

ϕε(x, x3) =∑k∈N

εkϕk(ε−1x, x3).

From the Ansatz, we deduce that

ϕ ∈ V := (ψi)i∈N such that for all x ∈ ω, we have ψ0(x,±L) = c± and ψ1(x,±L) = R±x

is such that for all ε,Jε(ϕ) = inf

ψ∈VJε(ψ),

whereJε(ψ) = Jε(ψε),

where ψε is defnied by

ψε(x, x3) =∑k∈N

εkψk(ε−1x, x3).

A simple change of variable gives us that

Jε(ψ) =

1

2

∫Ω

W

( ∞∑k=−1

εk(∇ψk, ∂3ψk−1)

)dx−

∫Ω

f(εx, x3)∑k

εkψk dx.

First, it can be easly check that the energy of the minimizers is uniformlybounded (it suffies to chose an element ψ ∈ V such that ψ0 depends only onx3). Assuming morevoer that W is infinite at infinity, we deduce that

∞∑k=−1

εk(∇ϕk+1, ∂3ϕ0)

is bounded and thus that ∇ϕ0 = 0, meaning that ϕ0(x, x3) is independant of x.We will use the notation ϕ0(x3) in place of ϕ0(x, x3) in the following. We now

set M−1 the subset of the elements ψ of V such that ∇ψ0 = 0. We have that

Jε(ϕ) ≤ Jε(ψ)

for all ψ ∈ M−1 and for all ε > 0. Letting ε goes to zero, we get that for allψ ∈M−1,

J(ϕ) ≤ J(ψ)

where

J(ψ) =1

2

∫Ω

W (∇ψ1, ∂3ψ0) dx−∫

Ω

f(0, x3)ψ0(x3) dx.

As J do only depend on ψ0 and ψ1 it can be minimize first with respect to ψ1

then with resepect to ψ0. For all ψ0 : (−L,L)→ R3 such that ψ0±L = ±c, weset

J(ψ0) = infψ1

1

2

∫Ω

W (∇ψ1, ∂3ψ0) dx−∫

Ω

f(0, x3)ψ0(x3) dx


and we haveJ(ϕ0) = inf J(ψ0).

It remains to get an explicit expression of J . The minimization with respect toψ1 can be made pointwise in x3, leading to

J(ψ0) = infψ1

1

2

∫ L

−L

(infψ1

∫ω

W (∇ψ1, ∂3ψ0) dx

)dx3 − |Ω|

∫ L

−Lf(0, x3)ψ0(x3) dx3.

For all v ∈ R3, we set

W (v) = |ω| infF∈R3×R2

W (F, v).

As W is infinite at infinity and assumed to be regular, the minimum with respectto F is reached. We denote F (x3) a minimizer for v = ∂3ψ0. Choosing

ψ1(x) = F (x3)x,

we get that

J(ψ0) =1

2

∫ L

−LW (∂3ψ0) dx3 −

∫ L

−Lf∗(x3)ψ0(x3) dx3, (5.3)

where f∗(x3) = |ω|f(0, x3).

5.2.2 Comments

The limit problem is for reasonable stored energy W not well posed. Indeed,a minimal requirement is for W to be frame indifferent, meaning that for allrotation R, we should have W (RF ) = W (F ) for all matrix F ∈ R3 ×R3. Fromthe definition of W , we deduce that it is also frame indifferent, that is

W (Rv) = W (v),

for all rotation R and for all vector v ∈ R3. Note that it implies in particularthat W do only depend on the |v|. Furthermore, if Ωε is the natural state ofthe body, we also have W (Id) = 0, so that W (1) = W (−1) = 0. We alreadyhave seen that a functional of the form (5.3) is weakly lower semicontinuous ifand only if W is convexe. The lower semi continuity is required to hope for theexistence of a minimizer of J . Thus, if the limit problem is well posed, we getthat W (v) = 0 for all |v| < 1 and in particular that W (0) = 0, that is

infF∈R3×R2

W (F, 0) = 0.

This is physically not reasonable. It would mean that you can flatten yourelastic body without using any energy. In the next section, we are going to seethat the correct limit is obtained by replacing W by its convexe envelope CWof W . But first, we will need some few theroretical tools.

Chapter 6

Γ-convergence

6.1 Definition and basic properties

Let X be a metric space and Fn a sequence of functiontals from X into R =R ∪ −∞ ∪ +∞.

Definition (Γ-convergence) A sequence Fn is said to be Γ-convergent to-ward F , if for every x ∈ X,

• F (x) ≤ lim inf Fn(xn) for all sequence xn converging toward x.

• There exists a sequence xn converging toward x such that

F (x) = limFn(xn).

Note that Γ-convergence is very different from pointwise convergence. For in-stance if Fn is a constant functional F , it Γ-converges toward the lower semi-continuous envelope of F (see next section). Some other usefull properties

Proposition 6.1.1. If Fn Γ-converge toward F , then F is lower semicontinu-ous.

Proof. Let x be an element of X and xn be a sequence converging toward x.We want to prove that

F (x) ≤ lim inf F (xn).

From the second property of the Γ convergence, we know that for all n, thereexsits a sequence xn,m converging toward xn such that

F (xn) = limmFm(xn,m)

It follows, that for all n, there exists M(n) (which can be assumed to beincreasing with respect to n) such that for all m ≥M(n), we have

|xn − xn,m| < 1/n, (6.1)

31

32 CHAPTER 6. Γ-CONVERGENCE

and|F (xn)− Fm(xn,m)| < 1/n. (6.2)

For all m ≥ M(0), there exists a unique nm such that M(nm) ≤ m <M(nm + 1). Note that nm is an increasing sequence with value in N. Moreover,it is onto and goes to infinity as m goes to infinity. We define the sequenceym = xnm,m for all m ≥ M(0). For m < M(0), we complete the sequence ymarbitrarily. We have

|ym − x| = |ym − xnm|+ |xnm

− x|.

It follows firstly that |xnm− x| goes to zero as m goes to infinity. Secondly, as

|ym − xnm| = |xnm,m − xnm

| < 1/nm

|ym−xnm| goes to zero when m goes to infinity. We conclude that ym converges

toward x as m goes to infinity.From the definition of the Γ-convergence, we get that

F (x) ≤ lim infm

Fm(ym) = lim infm

Fm(xn,nm),

and from (6.2),

F (x) ≤ lim infm

Fm(xn,nm) ≤ lim inf

m(F (xnm

) + 1/nm) = lim infm

F (xnm).

Finally, as mn is onto, we obtain that

F (x) ≤ lim infn

(F (xn).

Proposition 6.1.2. If Fn Γ-converge toward F and is G is a continuous map,then Fn +G does Γ converge toward F +G.

Exercise 7. Prove Proposition 6.1.2.

We admit the two following theorems, which will be used in the following.

Theorem 6.1.3 (Fundamental Theorem of Γ-convergence). Let Fn be a se-quence of functionals from X into R that Γ converges toward a functional F .Assume moreover, that the sequence Fn is equicoercive, meaning that there exitsa compact set K of X such that for all n,

infFn(x) : x ∈ X = infFn(x) : x ∈ K,

Then, the minima of Fn converge toward the minima of F , that is

minF (x) : x ∈ X = limn

infFn(x) : x ∈ X.

Moreover, if xn is a sequence of almost minimizers, that is if

Fn(xn) = inf Fn + rn,

with rn → 0, and if xn converge toward an element x ∈ X, then x is a minimizerof F .

6.1. DEFINITION AND BASIC PROPERTIES 33

Theorem 6.1.4 (Compactiy). Assume that X is separable (meaning that itadmits a countable dense subset), then every sequence Fn of function from Xinto R do admit a Γ-convergent subsequence.

An alternative characterization of the Γ convergence can be usefull

Proposition 6.1.5. A sequence of functionals Fn converges toward F if andonly if for all x ∈ X,

1. F (x) ≤ lim infn Fn(xn) for all sequence xn conveging toward x,

2. For all r > 0, F (x) ≥ lim supn→∞ infy∈B(x,r) Fn(y),

where B(x, r) is the ball of center x and radius r.

Proof. We begin to prove that any Γ converging sequence satifies the conditionsof the Proposition. Let Fn be a sequence that Γ converges toward a functionalF . We just have to prove the second statement of the Proposition. Let x ∈ X.From the definition of the Γ convergence, there exists a sequence xn → x suchthat

F (x) = limnF (xn).

For all r > 0, we have xn ∈ B(x, r) for n big enough, so that

infy∈B(x,r)

Fn(y) ≤ Fn(xn).

By taking the limsup in the previous inequality, we get

lim supn

infy∈B(x,r)

Fn(y) ≤ lim supn

Fn(xn) = F (x),

as claimed. Next, we have to establish the converse implication. Assume thatFn is a sequence of functionals as in the Proposition. Then, for all m > 0 andall n ≥ 0, there exists xn,m ∈ B(x, 1/m) such that

Fn(xn,m) ≤ infy∈B(x,1/m)

Fn(y) + 1/m.

From the second assumption on the sequence Fn, we get that

lim supn

Fn(xn,m) ≤ F (x) + 1/m.

Thus, fo every m > 0, there exists N(m) such that

Fn(xn,m) ≤ F (x) + 2/m,

for all n ≥ N(m). Moreover, without loss of generality, we can assume thatN(m) is strictly increasing. We set xn = xn,mn

where mn ∈ N is the onlyelement such thatN(mn) ≤ n < N(mn+1). Note thatmn is correctly defined as


long as n is greater or equal than N(1) (we can complete the sequence arbrtariltyfor n < N(1). We have

Fn(xn) = Fn(xn,m) ≤ F (x) + 2/mn.

As n + 1 < N(mn), we have that mn goes to infinity as n goes to infinity, sothat

lim supn

Fn(xn) ≤ F (x).

Finally, from the first assumption of the proposition, we get

F (x) ≥ lim supn

Fn(xn) ≥ lim infn

Fn(xn) ≥ F (x).

We conclude that all the inequalities in the previous equation are eqaulities. Inparticular lim supn Fn(xn) = lim infn Fn(xn) and that Fn(xn) is a convergingsequence. We thus have

F (x) = limnFn(xn),

and Fn is Γ converging toward F .

A last useful Lemma, which enable us to obtain the Γ-convergence of a wholesequence.

Lemma 6.1.6 (Independant Γ-limit). Let Fn be a sequence of functionals froma separable space X into R. Assume that there exsits F : X → R such that forevery Γ-converging subsequence of Fn do converges toward F . Then the wholesequence Fn does Γ-converge toward F .

Proof. Let us assume that Fn does not Γ converges toward F . Then eithercondition 1 or 2 of proposition 6.1.5 is false. Fristly, if condition 1 is false, thenthere exists x ∈ X and xn → x such that

F (x) > lim infn

Fn(xn).

Thus, there exists a strictly increasing map ϕ from N into N such that Fϕ(n)(xϕ(n))is convergent toward lim infn Fn(xn). Moreover, from the compacity Theorem,we can assume that Fϕ(n) is convergent. From the hypothesis of the Lemma,the Γ-limit of Fϕ(n) is F and we get

F (x) ≤ lim inf Fϕ(n)(xϕ(n)) = lim infn

Fn(xn) < F (x).

We conclude that the first condition 1 of Proposition 6.1.5 is true.Secondly, let us assume that the second condition of 2 is false. Then there

exists r > 0 such that

F (x) < lim supn

infy∈B(x,r

Fn(y).

6.2. LOWER SEMICONTIUOUS ENVELOPE 35

Once again, we can extract a subsequence such that Fϕ(n) does Γ-convergetoward F and such that

limn

infy∈B(x,r

Fϕ(n)(y) = lim supn

infy∈B(x,r)

Fn(y).

We get

F (x) < limn

infy∈B(x,r

Fϕ(n)(y) = lim supn

infy∈B(x,r)

Fϕ(n)(y) ≤ F (x).

This is once again contradictory, and the second condition of Proposition 6.1.5is verified. Finally, we conclude that the whole sequence Fn does Γ-convergetoward F as claimed.

To determine the behavior of the minimizers of a sequence of functionals,we proceed in three steps. First, by the compactness theorem, we can extracta Γ-converging subsequence. Next, we compute the Γ-limit and prove that itis independant of the chosen subsequence. Finally, we deduce from the Lemma6.1.6 that the whole sequence is Γ-convergent. The behavior of the minimizersis then given by the Fundamental Theorem of Γ-convergence.

6.2 Lower semicontiuous envelope

In the next section, we are going to apply Γ-convergence theory to prove computerigourously the behaviour of an elastic string. As underlined by the commentsof the previous chapter, the limit obtained formally is not correct. As we willsee, the correct limit is the lower semicontinuous envelope of the formal one.

Definition (Lower semicontinuous envelope) LetX be a topological spaceand J a map from X into R. The lower semicontinuous envelope (or relaxedfunction) sc- J of J is the greatest semi continuous map lower than J .

Note that, the surpremum of lower semicontinuous functions being lower semi-continuous, the semicontinuous envelope is always correctly defined. The lowersemicontious envelope gives us a description of the behavior of the minimizationsequences of J .

Theorem 6.2.1. Let X be a metric space, and if the minimization sequencesof J remain in a compact subset of X, then

1. The minimization sequences of sc- J remain in a compact subset of X andsc- J is lower semicontinuous.

2. The function sc- J admits a minimizer on X.

3.

minX

sc- J = infXJ.


4. If xn is a minimization sequence of J converging toward an element x ∈ X,then x is a minimizer of sc- J .

5. If x is a minimizer of sc- J , then there exists a minimization sequence ofJ converging toward x.

Exercise 8. Prove the Theorem 6.2.1.

6.2.1 Relaxation in the one dimensional case

Let us consider a map W ∈ C(Rd,R) (with d ∈ N∗) such that there exists1 < p <∞, such that

∀F ∈ Rd, C0|F |p − C1 ≤W (F ) ≤ C2(|F |p + 1), (6.3)

where C0, C1 and C2 are postive constants. We introdcue the functional J overW 1,p(I)d, where I = (0, 1) defined by

J(ψ) =

∫I

W (∇ψ) dx. (6.4)

Definition (Convex envelope) The convex envelope CW of a map W is thegreatest convex function lower than W .

Proposition 6.2.2. The lower semi continuous envelope of J defined by (6.4)for the Lp norm is defined by

sc- J(ψ) =

∫I

CW (∇ψ) dx.

To prove this proposition, it is useful to use an alternative definition of theconvex envelope of W .

Proposition 6.2.3. Let W ∈ C(Rd,R). Then

CW (F ) = inf

∫I

W (F +∇ϕ) dx : ϕ ∈W 1,p0 (I)

Proof. First, we have from the Jensen’s inequality

CW (F ) = CW

(∫I

F +∇ϕ)≤∫I

CW (F +∇ϕ) dx ≤∫I

W (F +∇ϕ) dx.

It only remain to prove that

CW (F ) ≥ inf

∫I

W (F +∇ϕ) dx : ϕ ∈W 1,p0 (I)

To this end, it suffies to prove that the map QW defined by

QW (F ) = inf

∫I

W (F +∇ϕ) dx : ϕ ∈W 1,p0 (I)


is convex. Let F1 and F2 be elements of Rd and θ ∈ (0, 1), we have

QW (θF1 + (1− θ)F2) = inf

∫I

W (θF1 + (1− θ)F2 +∇ϕ) dx : ϕ ∈W 1,p0 (I)

Now for all map ϕ1 ∈ W 1,p

0 (0, θ) and ϕ2 ∈ W 1,p0 (θ, 1), we define ϕ ∈ W 1,p

0 (I)such that x ∈ (0, θ),

θF1 + (1− θ)F2 +∇ϕ(x) = F1 +∇ϕ1(x)

and for all x ∈ (θ, 1),

θF1 + (1− θ)F2 +∇ϕ(x) = F1 +∇ϕ2(x).

Note that there exsits indeed such a map (and it is unique). To this end, wejust have to check that the integral over I of ∇ϕ is equal to zero. Thus, weobtain that

QW (θF1 + (1− θ)F2)

≤ inf

∫ θ

0

W (F1 +∇ϕ1) dx+

∫ 1

θ

W (F2 +∇ϕ2) dx :

ϕ1 ∈W 1,p0 (0, θ) and ϕ2 ∈W 1,p

0 (θ, 1)

= inf

∫ θ

0

W (F1 +∇ϕ1) dx : ϕ1 ∈W 1,p0 (0, θ)

+ inf

∫ 1

θ

W (F2 +∇ϕ2) dx : ϕ2 ∈W 1,p0 (θ, 1)

.

After a simple change of variable, we get that

QW (θF1 + (1− θF2) ≤ θQW (F1) + (1θ)QW (F2),

and that QW is indeed convex.

Proposition 6.2.4 (Jensen’s Inequality). Let J be a convex function from Rdinto R, and Ω a set of bounded measure, then for all u ∈ L1(Ω)d, we have

J

(|Ω|−1

∫Ω

u(x) dx

)≤ |Ω|−1

∫Ω

J(u(x)) dx.

Proof. Let t = |Ω|−1∫

Ωu(x) dx. As J is convex, there exists a linear map L on

Rd such thatJ(s) ≥ J(t) + L(s− t),

for all s ∈ Rd. In particular,

J(u(x)) ≥ J(t) + L(u(s)− u(x)),


By intergration, we get ∫Ω

J(u(x)) dx ≥ |Ω|J(t).

Proof of Proposition 6.2.2. Firstly, we obviously have for all ϕ ∈W 1,p(Ω)∫Ω

CW (∇ϕ) dx ≤∫

Ω

W (∇ϕ) dx.

Moreover, the left hand side is lower semi continuous for the Lp(I)d topology.Indeed, let ϕn be a converging sequence toward an element ϕ ∈ Lp, such that∫

ΩCW (∇ϕn) dx is bounded. As CW statisfies similar growth assumptions that

W , the sequence ϕn is bounded in W 1,p(I)d. Thus it is converging toward ϕ forthe weak topology of W 1,p(I)d. As the left hand side is lower semicontinuousfor the weak topology of W 1,p(I)d, we get that∫

Ω

CW (∇ϕ) dx ≤ sc- J(ϕ).

It remains to prove the converse inequality. First let us consider a linear map ϕ,and let ψ ∈ W 1,p

0 (I), for all integer n, let ψn be the 1/n-periodic map definedon (0, 1/n) by

ψn(x) = ψ(nx)/n.

We set ϕn = ϕ+ ψn. Obviously, ϕn do converge toward ϕ for the Lp topology.Thus,

sc- J(ϕ) ≤ lim inf sc- J(ϕn) ≤ lim inf J(ϕn) = lim inf

∫I

W (∇ϕn) dx.

After a change of variable, we get that

sc- J(ϕ) ≤∫I

W (∇ϕn) dx =

∫I

W (∇ϕ+∇ψ) dx.

Taking the infimum over the maps ψ ∈W 1,p0 (I), we get that

sc- J(ϕ) ≤∫I

CW (∇ϕ) dx.

Next, it is easy to extend this result to any ϕ affine by parts (we only have toapply the previous argument to each interval where ϕ is affine). Finally, for anyϕ ∈W 1,p(I)d, there exists a sequence ϕn of maps affine by parts that convergestoward ϕ for the strong W 1,p topology. We have

sc- J(ϕ) ≤ lim infn

sc- J(ϕn) ≤ lim inf

∫I

CW (∇ϕn) dx =

∫I

CW (∇ϕ) dx.


6.2.2 Relaxation in the scalar case

Let W ∈ C(Rn,R), such that there exists postive constants C0, C1 and C2 suchthat for all F ∈ Rn we have

C0|F |p − C1 ≤W (F ) ≤ C2(|F |p + 1),

with 1 < p <∞. Let Ω be a bounded open subset of Rn and let J : W 1,p(Ω)→ Rdefined by

J(u) =

∫I

W (∇u) dx. (6.5)

Proposition 6.2.5. The lower semi continuous envelope of J defined by (6.5)for the Lp norm is defined by

sc- J(u) =

∫Ω

CW (∇u) dx.

The proof is very similar to the one performed in the one dimensional case.

6.2.3 Relaxation in the vectorial case

We consider this time a functional J ∈ W 1,p(Ω;Rd), where Ω is an boundedopen subset of Rn defined by

J(ϕ) =

∫Ω

W (∇ϕ) dx. (6.6)

Once again, we assume, that W is a continuous map and that there existspositive constants C0, C1 and C2 such that

C0|F |p − C1 ≤W (F ) ≤ C2(|F |p + 1).

Definition (Definition) Let W ∈ C(Rn×d,R). The quasiconvex envelope ofW is defined by

QW (F ) = inf

|D|−1

∫D

W (F +∇ϕ) with ϕ ∈W 1,∞0 (Ω)

.

We give a relaxation result for J without proof (the proof, which is a lotmore complicated than in the scalar or one dimensional cases can be found in[?])

Proposition 6.2.6. Let J defined by (6.6), then the lower semicontinuous en-velope of J for the Lp toplogy is given by

sc- J(ϕ) =

∫Ω

QW (∇ϕ) dx.


6.3 Application to elastic rods

We consider once again the problem consisting in determining the behavior ofthe minimizers of an elastic rod. We use the same notations than the one usedin the formal derivation. Moreover, we assume that W is continuous and thatthere exists p ∈ (1,∞) and positive constants C1, C2 and C3, such that

C1|F |p − C2 ≤W (F ) ≤ C3(|F |p + 1) for all F ∈ R3×3.

We also assume that f is independant of x and that f ∈ Lp′(−L,L)N and weset

J(ε)(ψ) =1

2

∫Ω

W (ε−1∇ψ, ∂3ψ) dx−∫

Ω

f(x3)ψ(x) dx.

The minimization problem of Jε is equivalent to the minimization problem ofJ(ε) over

Vε = ψ ∈W 1,p(Ω)N : ψ(x,±L) = εR±x+ c±.

We can not apply directly our Γ convergence result to J(ε). First because J(ε)is define over a set Vε that does depend on ε (due to the limit conditions) andsecondly, because we can not hope the squence J(ε) to be equicoercive for thenorm W 1,p. To avoid both of those problems, we extend J(ε) to Lp(Ω)3 settingJ(ε)(ψ) =∞ for all v ∈ Lp(Ω)3 that do not belong to Vε.

Proposition 6.3.1. The sequence J(ε) Γ-converge toward the functional J fromLp(Ω)3 into R defined by

J(ψ) =1

2

∫ L

−LCW (∂3ψ) dx3 −

∫ L

−Lf∗ψ dx3

for all

ψ ∈ V := ψ ∈ Lp(Ω)3 such that

there existe ψ0 ∈W 1,p(−L,L) such that ψ(x) = ψ0(x3) and ψ0(±L) = ±c,

and J(ψ) =∞ for all ψ ∈ Lp(Ω)3 \ V .

Proof. First, from the compacity Theorem of Γ-convergence, there exists a sub-sequence (still denoted J(ε)) and a functional J such that J(ε) do Γ-convergetoward J . We propose to identify J and prove that it is independant of thechosen subsequence. Thus, will we obtain the full convergence of the sequenceJ(ε) from Lemme 6.1.6.

In fact, as the second part in the expression of the energy is continuous andindependant of ε, we can focus, by Proposition 6.1.2, on the computation of theΓ-limit of the first part of the energy, that is

I(ε)(ψ) =1

2

∫Ω

W (ε−1∇ψ, ∂3ψ) dx.

We denote by I its Γ-limit.

6.3. APPLICATION TO ELASTIC RODS 41

Lower bound for the I In this section, we want to prove that for all ψ ∈ V

1

2

∫ L

−LCW (∂3ψ) dx3 ≤ I(ψ)

and that for all ψ /∈ V , I(ψ) =∞.Let ψ ∈ Lp(Ω)3 such that I(ψ) <∞. Then, there exists ψε ∈ Lp(Ω)N such

thatI(ψ) = lim inf

εI(ε)(ψε) <∞.

Form the growing property of W , be conclude that∫Ω

|ε−1∇ψε, ∂3ψε|p ≤ C∫

Ω

W (ε−1∇ψε, ∂3ψε) dx <∞

In particular, ∇ψε is bounded in Lp(Ω)3 and from the Poincare inequality, wededuce that ψε is bounded in W 1,p(Ω)3. Thus, it admits a converging subse-quence in W 1,p(Ω)3 for the weak topology. As its limit is independant of thechosen subsequence, it follows that the whole sequence ψε is weakly convergentin W 1,p(Ω)3. Moreover, we have that ∇ψε strongly converges in Lp(Ω)3×2 to-ward zero. Thus, ψ belongs W 1,p(Ω)3 and is independant of x. Moreover, thetrace being continuous for the weak topology, we deduce from

ψε(x,±L) = εR±x+ c±

thatψε(±L) = c±.

It follows that ψ bleongs to V . We thus have obtained that for all ψ /∈ V ,I(ψ) =∞. Next, we have

1

2|ω|

∫Ω

CW (∂3ψε) dx ≤ 1

2|ω|

∫Ω

W (∂3ψε) dx

≤ 1

2

∫Ω

W (ε−1∇ψε, ∂3ψ − ε) dx

= I(ε)(ψε).

The left hand side of this inequality is lower semicontinuous for the weakW 1,p(Ω)3 topology (it is a convex continuous in W 1,p). Thus, taking the limitinf in this inequality, we get that

1

2|ω|

∫Ω

CW (∂3ψ) dx ≤ I(ψ)

and as ψ is indepenant of x, we get

1

2

∫ L

−LCW (∂3ψ) dx3 ≤ I(ψ).


Upper bound for the I We are going to prove that for all ψ ∈ V ,

I(ψ) ≤ 1

2

∫ L

−LCW (∂3ψ) dx3.

In fact, we only to prove that

I(ψ) ≤ 1

2

∫ L

−LW (∂3ψ) dx3.

As I(ψ) is lower semicontinuous for the norm Lp(Ω)3, it is semicontinuous for theweak topology of W 1,p(Ω)3. So, I(ψ) is lower that the semicontinuous envelopeof the right hand side, which is precisely obtained by taking the convex envelopeof W . Next, let us define

ψε(x, x3) = ψ(x3) + εF (x3)x,

where F is a map in W 1,p(−L,L)3×2 such that F (±L) = ±R. Obviously,ψε ∈ Vε do converge toward ψ in Lp(Ω)3, thus

I(ψ) ≤ lim infε

I(ε)(ψε) =1

2

∫Ω

W (ε−1∇ψε, ∂3ψε) dx.

Moreover,∇ψε = εF.

and∂3ψε = ∂3ψ + ε∂3Fx.

We get using the Lebesgue dominated convergence theorem that

I(ψ) ≤ lim infε

1

2

∫Ω

W (F (x3), ∂3ψ + ε∂3F (x3)x) dx

=1

2

∫Ω

W (F, ∂3ψ) dx.

We thus have

I(ψ) ≤ infF∈W 1,p(−L,L)3×2

|ω|2

∫ L

−LW (F, ∂3ψ) dx. (6.7)

To conclude, let F a map from (−L,L) into R3×2 such that

|ω|W (F , ∂3ψ) = W (∂3ψ).

We have

C1|F |p − C2 ≤W (F , ∂3ψ) ≤W (0, ∂3ψ) ≤ C3(|∂3ψ|p + 1).

It follows that F belongs to Lp(−L,L)3×2. Finally, the set of maps such thatF in W 1,p(−L,L)3×2 such that F (±L) = ±R being dense in Lp(−L,L)3×2, weconclude from (6.7) that

I(ψ) ≤ 1

2

∫ L

−LW (∂3ψ) dx.

6.4. APPLICATION TO ELASTIC MEMBRANES 43

6.4 Application to elastic membranes

In this section, we are going to consider thin elastic membranes. For all ε > 0,we set Ωε = ω × (−ε, ε), where ω is a bounded open subset of Rn−1. Weassume that Ωε is made of an hyperelastic material, partial embended on a partof its boundary Γε = γ × (−ε, ε) and submitted to volumic dead body loadsf ∈ Lp(Ω;Rd). The energy of a deformation ϕ ∈W 1,p(Ω;Rd) is given by

Jε(ϕ) =1

2

∫Ω

W (∇ϕ) dx−∫

Ω

f · ϕdx.


Chapter 7

Other examples (formalapproach)

7.1 Elastic beams

7.2 Elastic plates

7.3 Modica Mortola

45

46 CHAPTER 7. OTHER EXAMPLES (FORMAL APPROACH)

Bibliography

[1] Gregoire Allaire. Homogenization and two-scale convergence. SIAM J. Math.Anal., 23(6):1482–1518, 1992.

[2] Andrea Braides. Introduction to homogenization and gamma-convergence.

[3] Haım Brezis. Analyse fonctionnelle. Collection Mathematiques Appliqueespour la Maıtrise. [Collection of Applied Mathematics for the Master’s De-gree]. Masson, Paris, 1983. Theorie et applications. [Theory and applica-tions].

[4] Gianni Dal Maso. An introduction to Γ-convergence. Progress in NonlinearDifferential Equations and their Applications, 8. Birkhauser Boston Inc.,Boston, MA, 1993.

[5] Herve Le Dret and Annie Raoult. Le modele de membrane non lineairecomme limite variationnelle de l’elasticite non lineaire tridimensionnelle. C.R. Acad. Sci. Paris Ser. I Math., 317(2):221–226, 1993.

[6] Herve Le Dret and Annie Raoult. The nonlinear membrane model as vari-ational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl.(9), 74(6):549–578, 1995.

[7] Walter Rudin. Analyse reelle et complexe. Masson, Paris, 1980. Trans-lated from the first English edition by N. Dhombres and F. Hoffman, Thirdprinting.

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