PietroSiorpaesProf.LuigiAmbrosio Homogenizationofintegralfunctionals

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Universit`a di Pisa

Facolt`a di Scienze Matematiche Fisiche e Naturali Corso di Laurea in Matematica

A. A. 2005-06

Tesi di laurea

23 giugno 2006

Homogenization

of integral

functionals

Candidato

Relatore

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Chapter 1 Introduction

In homogenization theory one studies the asymptotic behavior of fastly oscil-lating systems of partial differential equations or of integral functionals, de-pending on a scale parameter ε which tends to zero. In this work we study the homogenization of integral functionals, that is the behavior of families (Fε)ε>0

of the type Fε(u) = Z Ω fx ε, Du(x) dx , (1.1)

with Ω bounded open set of Rn _{and u ∈ W}1,p_{(Ω; R}m_). _{We will see that}

these families converge in some sense to a functional of the type F (u) = R

Ωfhom(Du(x))dx and that the minimizers uε of Fε converge to the minimizer

u of F . In this framework the major tasks of homogenization theory are the proof of this convergence, the description of fhom as a solution of an auxiliary

problem, and the study of its qualitative and quantitative properties such as convexity, coerciveness, smoothness.

Since the minimizers uε are in general wildly oscillating, their convergence

to u must be understood in the weak sense of Sobolev spaces. The natural framework for the study of this problem is then the setting of weakly lower semicontinuous functionals on W1,p_{(Ω; R}m_{), so the notions of quasiconvexity}

and of W1,p_{(Ω; R}m)-quasiconvexity turn out to be important. We shall state the basic facts about these and other notions of convexity (namely polycon-vexity and rank-one conpolycon-vexity) in the preliminary results.

Γ-convergence has proved to be a very important notion of convergence in the class of lower semicontinuous functionals and implies, under suitable equi-coercivity assumptions, the convergence of the minimizers uε to the minimizer

u. In particular Γ-convergence can be used to attack very general problems of homogenization, in a non linear setting and with minimal regularity hypothe-ses.

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CHAPTER 1. INTRODUCTION

We apply the direct methods of Γ-convergence, which consist in a localization procedure that highlights the dependence of integrals as set functions, so in our case we consider Fε(u, U ) =

R

Ufε(x, Du(x))dx with U open ⊆ Ω.

Next, general abtract Γ-compactness results apply, so given a sequence εj → 0+

we obtain the existence of a subsequence εjk along which there exists the

Γ-limit F (u, R) of Fε_jk(u, R) at any u ∈ W1,p(Ω; Rm) and for every R belonging

to a countable basis of Ω. A representation theorem then can be applied if we prove that this holds also for every U open subset of Ω and that F (u, ·) is the trace of a Borel measure. To do this, it suffices to show that the Γ-lim inf and the Γ-lim sup are, at every u ∈ W1,p_{(Ω; R}m_{), inner regular increasing set}

functions, and that the Γ-lim sup is subadditive.

The main technical ingredient is the so-called “Lp-fundamental estimate”, whose aim is to “join functions without introducing a large error”. In the case of functionals of the type Fε(u, U ) =

R

Ufε(x, Du(x))dx, it asserts that

under various growth conditions for fε, given three open subsets U0, U, V of Ω

with U0 _{b U and two functions u, v in W}1,p_{(Ω; R}m_{), for sufficient small ε it is}

possible to find a suitable interpolation w between u and v, coinciding with u on U0 and with v on V \ U , and such that

Fε(w, U0∪ V ) ≤ Fε(u, U ) + Fε(v, V ) + C

Z

U ∩V

|u − v|pdx (with C positive constant) up to an error which vanishes as ε → 0.

In chapter (6) we consider the case of standard growth conditions of order p, i.e. the coercive case where α|A|p ≤ fε(x, A) ≤ β(1 + |A|p), and the case of

the “doubling” hypothesis, where we suppose that there exists a function g such that g(x, A) ≤ fε(x, A) ≤ C(1 + g(x, A)), g(x, 2A) ≤ C(1 + g(x, A)) and

0 ≤ g(x, A) ≤ β(1+|A|p). In chapter (8) we condider the non-standard growth conditions, that is α|A|p _{≤ f}

ε(x, A) ≤ β(1 + |A|q), where p ≤ q < p∗.

By the representation theorem we obtain that F (u, U ) =R_Uφ(x, Du(x)) for a suitable quasiconvex function φ. In the case of homogenization, i.e. fε(x, A) =

f (x/ε, A), if f is 1-periodic in x and satisfies the standard growth condition, we prove that φ(x, A) is indipendent from x and from the subsequence εjk,

thus not only (Fε_jk) but the whole family (Fε) Γ-converges to

R

Uφ(x, Du(x)).

To do this the crucial point is the proof of the existence of the limit lim t→+∞inf 1 tn Z (0,t)n f (x, A + Du(x))dx : u ∈ W₀1,p((0, t)n_{; R}m) , (1.2) which defines fhom(A) and which is called the asymptotic homogenization

for-mula. Then, being φ W1,p_{-quasiconvex, φ(A) equals}

min Z (0,1)n φ(A + Du(x))dx : u ∈ W₀1,p((0, 1)n_{; R}m) . (1.3)

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From the Γ-convergence property of convergence of infima to minima, after a change of variables we deduce that the limit in (1.2) equals (1.3), so φ equals fhom and so is indipendent from the subsequence εjk.

In the convex and in the scalar cases a simpler description can be given, which takes into account only one minimum problem: the so-called cell-problem for-mula holds

fhom(A) = min

Z

(0,1)n

f (x, A + Du(x))dx : u ∈ W_#1,p((0, 1)n_{; R}m)

, (1.4)

where W_#1,p((0, 1)n_{; R}m_{) is the set of all 1-periodic functions in W}1,p loc(R

n_{; R}m_).

Moreover, in the convex case we can obtain the same homogenization results just under the “doubling” hypothesis, so without any coercivity assumption. Then we extend the above homogenization results to the “almost periodic” case. It applies in particular to the cases where we consider integrands of the form f (x, A) + g(x, A), with f and g periodic but without a common period, or of the form f (x, s, A), where, even though f is periodic in the first two variables, the function x 7→ f (x, Ax, A) need not be periodic. Here we consider functionals of the type

Fε(u, U ) = Z U fx ε, u ε, Du(x) dx , (1.5)

with f (x, s, A) which satisfies conditions similar to Bohr almost periodicity, but with a sort of “uniformity” which takes into account the dipendence of f from the other variables. To do this we work with almost periods instead of periods, without major changes.

Then we apply the preeceding theorems to homogenize elliptic equations, find-ing a representation of the limit involvfind-ing solutions of an Euler-Lagrange equa-tion, and to a riemannian metric, where we work out an example that shows explicitely that the Γ-limit integral of functionals of the form

Fε(u) = Z 1 0 n X i,j=1 au ε u0_iu0_jdt (1.6)

is a functional of the form

F (u) = Z 1

0

φ(u0) dt , (1.7)

where φ need not be a quadratic form.

The “almost periodic” case can be used to obtain homogenization results un-der very mild “regularity” assumptions on the integrand, namely unun-der the

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CHAPTER 1. INTRODUCTION

assumption that f (·, A) is Besicovitch almost periodic, so in particular for any L1 _{function. To prove this we need the Yosida transform and a regularity}

theorem, which are used to prove that “homogenization is stable under small perturbations”. In particular the “homogenization closure theorem” states that if (fj) is a sequence of “homogenizable” integrands and if

lim j lim sup_{T →+∞} 1 (2T )n Z (−T,T )n sup |A|≤M |fj(x, A) − f (x, A)|dx = 0 (1.8)

for any M ≥ 0, then also f is “homogenizable” and (fj)hom converges to

fhom. Finally we generalize some results to the case of “multiple scales of

homogenization”, that is the case where Fε(u, Ω) = Z Ω fx,x ε, x ε2, . . . , x εk, Du(x) dx . (1.9)

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Chapter 2 Preliminary results

2.1 Notions of convexity

Considering functionals of the form F (u) =

Z

Ω

f (x, u, Du)dx , (2.1)

if we want continuity properties for F a natural condition for f is the following one. Take Ω open set of Rn, we say that f : Ω × Rm × Rm×n

→ R is a Carath´eodory function if f (x, ·, ·) is continuous for a.e. x ∈ Ω and f (·, s, A) is measurable for all (s, A) ∈ Rm_{× R}m×n_.

Given f : Ω×Rm×Rm×n

→ R, we will improperly say that f is a Borel function if there exists a Borel measurable function g such that f (x, s, A) = g(x, s, A) for a.e. x ∈ Ω and for every (s, A) ∈ Rm _{× R}m×n_{, or equivalently if f is}

L(Ω) × B(Rm

) × B(Rm×n) measurable. We remark that, with this definition, every Carath´eodory function is a Borel function.

Remark 2.1.1 If |f (x, s, A)| ≤ c(a(x) + |s|p_{+ |A|}p_{) for a.e. x ∈ Ω and for}

every (s, A) ∈ Rm _{× R}m×n _{and with a ∈ L}1_{(Ω), and if f is a Carath´}_eodory

function, then the functional F given by (2.1) is continuous with respect to the strong W1,p_{(Ω; R}m_{) convergence.}

We are interested in necessary or sufficient conditions on f in order to have that F is weakly lower semicontinuous in some Sobolev space W1,p_{(Ω; R}m_{). We}

will just need the simpler case where f : Rm×n _{→ [0, +∞] is a Borel function.}

We shall begin with the following definition, where p ∈ [1, +∞]. A reference for what follows is the original paper [4].

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CHAPTER 2. PRELIMINARY RESULTS

set E ⊂ Rn _{with |∂E| = 0 and for any φ ∈ W}1,p

0 (E; Rm) we have that

|E|f (A) ≤ Z E f (A + Dφ(x))dx , (2.2) or equivalently that f (A) = min 1 |E| Z E f (A + Dφ(x))dx : φ ∈ W₀1,p_{(E; R}m) . (2.3) f is W1,p_{-quasiconvex if it is W}1,p_{-quasiconvex at every A ∈ R}m×n_.

We remark that, to prove that f is W1,p_{-quasiconvex at A, it is sufficient to}

prove (2.2) for just one open bounded set E ⊂ Rn _{with |∂E| = 0. Moreover,}

if f takes only finite values, the condition |∂E| = 0 can be removed. The following relations hold beetween the different notions of W1,p_{-quasiconvexity.}

Remark 2.1.3 1. If 1 ≤ p ≤ q ≤ +∞, then W1,p-quasiconvexity implies W1,q_{-quasiconvexity.}

2. If 1 ≤ p < +∞, f is continuous and

0 ≤ f (A) ≤ c(1 + |A|p) _{∀A ∈ R}m×n (2.4) then f is W1,p_{-quasiconvex if and only if it is W}1,∞_{-quasiconvex, because}

of remark (2.1.1).

3. If 1 ≤ p < +∞, and f satisfies

c|A|p ≤ f (A) _{∀A ∈ R}m×n _(2.5)

then f is W1,p-quasiconvex if and only if it is W1,1-quasiconvex

The notion of W1,p_{-quasiconvexity is important because it is a necessary}

con-dition to have lower semicontinuity of the integral functional. Theorem 2.1.4 If F : W1,p_{(Ω; R}m_{) → [0, +∞] given by}

F (u) = Z

Ω

f (Du(x))dx (2.6)

is sequentially weakly lower semicontinuous (weakly∗ if p = +∞), where Ω ⊂ Rn is bounded and open, then f is lower semicontinuous and for every n-cube Y we have f 1 |Y | Z Y Dv(x)dx ≤ 1 |Y | Z Y f (Dv(x))dx (2.7)

for all v ∈ W1,p_loc _(Rn_{; R}m_{) with Dv Y -periodic.} _{In particular f is W}1,p

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2.1. NOTIONS OF CONVEXITY

We remark that condition (2.7) is in general strictly stronger then W1,p

-quasiconvexity.

Under proper growth conditions, W1,p-quasiconvexity is equivalent to Morrey’s quasiconvexity, and turns out to be a sufficient condition for lower semiconti-nuity.

Definition 2.1.5 A continuous function f : Rm×n → R is quasiconvex at A ∈ Rm×nif, for if for any open bounded set E ⊂ Rnand for any φ ∈ W01,∞(E; Rm),

we have that

|E|f (A) ≤ Z

E

f (A + Dφ(x))dx . (2.8)

f is quasiconvex if it is quasiconvex at every A ∈ Rm×n_.

We remark that, to prove that f is quasiconvex, it is sufficient to prove (2.8) for just one open bounded set E ⊂ Rn_{and for any φ ∈ C}∞

0 (E; Rm). A positive

quasiconvex function f is a W1,∞-quasiconvex positive continuous function which takes only finite values. In particular, thanks to remark (2.1.3.(2)), we have

Remark 2.1.6 If 1 ≤ p < +∞, f is continuous and

0 ≤ f (A) ≤ c(1 + |A|p) _{∀A ∈ R}m×n (2.9) then f is W1,p_{-quasiconvex if and only if it is quasiconvex.}

A generalization of this theorem can be found in [1].

Theorem 2.1.7 If Ω ⊂ Rn _{is open, 1 ≤ p < +∞ and f : R}m×n _{→ R is a}

quasiconvex function such that

0 ≤ f (A) ≤ c(1 + |A|p) _{∀A ∈ R}m×n , (2.10) then F : W1,p_{(Ω; R}m_{) → [0, +∞) given by}

F (u) = Z

Ω

f (Du(x))dx (2.11)

is sequentially weakly lower semicontinuous.

The quasiconvexity condition is somewhat difficult and technical to handle, so it is interesting to look for simpler properties related to quasiconvexity.

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Definition 2.1.8 A function f : Rm×n _{→ (−∞, +∞] is polyconvex if there}

exists a convex function g : Rτ (n,m) _{→ (−∞, +∞] such that}

f (A) = g(M (A)) _{∀A ∈ R}m×n , (2.12) where M (A) is the vector of all the minors of A and τ (n, m) is the dimension of the vector space of all the M (A) with A ∈ Rm×n_.

The study of polyconvex functions is carried out in [12]. For example the function f (A) = |A|p_{+ det(A) is polyconvex if 1 ≤ p < +∞. Notice that this}

function is not W1,p_{-quasiconvex if 1 ≤ p < 2. We remark that}

|Br(0)|M (A) =

Z

Br(0)

M (A + Dφ(x))dx (2.13)

for every φ ∈ C₀∞(Br(0); Rm), because every minor can be written as a

diver-gence and soR_B

r(0)M (A + Dφ(x))dx depends only on the boundary values of

Ax + φ(x), so we can evaluate the integral with φ = 0. From this and Jensen inequality it follows that

Remark 2.1.9 If f : Rm×n _{→ R is continuous and polyconvex, then it is}

quasiconvex.

Definition 2.1.10 f : Rm×n _{→ (−∞, +∞] is rank-one convex if for every}

A ∈ Rm×n, a ∈ Rm, b ∈ Rn the function t 7→ f (A + t a ⊗ b) is convex, i.e. if for any 0 ≤ t ≤ 1 we have

f (A + t a ⊗ b) ≤ tf (A + a ⊗ b) + (1 − t)f (A). (2.14) We will need the following remark.

Remark 2.1.11 If f is rank-one convex and |f (A)| ≤ c(1 + |A|p_{), then f}

satisfies the local Lipschitz condition

|f (A) − f (B)| ≤ c(1 + |A|p−1_{+ |B|}p−1_{)|A − B|} _(2.15)

for all A, B ∈ Rm×n_.

We remark that there exist lower semicontinuous positive functions which are W1,1_{-quasiconvex but are not rank-one convex. However if f is positive and}

locally bounded, then rank-one convexity follows from W1,∞_{-quasiconvexity.}

Moreover we have that quasiconvexity always implies rank-one convexity, and the contrary is not true (see [22]).

We have the scheme

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2.2. THE YOSIDA TRANSFORM

where in general all the inclusions are strict, but in the case n = 1 or m = 1 they are all equalities. It is clear that, if fi are convex, polyconvex or rank-one

convex, then so it is sup_ifi. So we can define the convex (polyconvex,

rank-one convex) envelope of f to be the max of all the g ≤ f which are convex (polyconvex, rank-one convex). However it is not at all obvious that a similar existence result holds for the quasiconvex envelope.

Definition 2.1.12 Let f : Rm×n _{→ R be a Borel function and E ⊂ R}n _{be a}

bounded open set with |∂E| = 0. We define the quasiconvexification Qf of f by Qf (A) = inf 1 |E| Z E f (A + Dφ(x))dx : φ ∈ C₀∞_{(E; R}m) (2.17) for every A ∈ Rm×n.

The definition (2.17) does not depend on the set E. The following result holds. Theorem 2.1.13 If f : Rm×n → R is a locally bounded Borel function then Qf is the quasiconvex envelope of f , that is

Qf = maxg : Rm×n _{→ R : g ≤ f and g is quasiconvex} . (2.18) Proofs of the properties of quasiconvexifications can be found in [12]. We will say that f : Ω × Rm× Rm×n

→ R is a quasiconvex function if, for a.e. x ∈ Ω and for any s ∈ Rm_{, A 7→ f (x, s, A) is quasiconvex.}

2.2 The Yosida transform

In chapter (7) we will need Yosida transform, which is useful as it permits to approximate a lower semicontinuous function with Lipschitz continuous func-tions.

Definition 2.2.1 Let (X, d) be a metric space and ψ : [0, +∞) → [0, +∞) be a strictly increasing continuous function with ψ(0) = 0. Given λ ≥ 0 and f : X → R, we define the Yosida transform of f , Tλψf : X → R, by

T_λψf (x) = inf{f (y) + λψ(d(x, y)) : y ∈ X}. (2.19) If ψ(t) = tp _{we write simply T}p

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Given a function f : X → R, f will denote the lower semicontinuous envelope of f , given by

f = max{g : g is lower semicontinuous and g ≤ f } = f (x) ∧ lim inf

y→x f (y) ,

and we will say that f is proper if −∞ < f (x) for any x ∈ X and f is not the constant +∞.

The Yosida transform enjoys the following useful properties.

Theorem 2.2.2 In the previous hyphothesis T_λψf (x) is increasing in λ and in f and T₀ψf = inf f ≤ T_λψ_{f ≤ f . Moreover if f : X → R is bounded from below} and proper, then

f = sup

λ≥0

T_λψf (2.20)

and for p ∈ (0, 1] we have that T_λp_{f is the maximum function g : X → R such} that g ≤ f and

|g(x) − g(y)| ≤ λd(x, y)p _. _(2.21)

2.3 Sobolev Spaces

Let us state two well known theorems that we will need many times. Canonical references for Sobolev spaces are [3] and [25].

Theorem 2.3.1 (Rellich’s compactness theorem) Take a bounded open set with Lipschitz boundary Ω ⊂ Rn, 1 ≤ p < n and p ≤ q < p∗ or n ≤ p < +∞ and p ≤ q < +∞. Then W1,p_{(Ω; R}m_{) ,→ L}q_{(Ω; R}m_{) with compact immersion.}

In particular, given F : W1,p_{(Ω; R}m_{) → R, if F is lower semicontinuous with}

respect to the strong Lp _{topology, then F is sequentially lower semicontinuous}

with respect to the weak W1,p topology.

If p > 1, F is sequentially lower semicontinuous with respect to the weak W1,p

topology and satsfies a growth condition of the type α

Z

Ω

|Du|p_{dx ≤ F (u) ,} _(2.22)

then F is lower semicontinuous with respect to the strong Lp _topology.

Given an open set U ⊂ Rn, we will say that φ : U → Rm is a piecewise affine function if there exist A1, . . . , Ak ∈ Rm×n, b1, . . . , bk ∈ Rm, and open disjoint

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2.3. SOBOLEV SPACES

Theorem 2.3.2 Let Ω ⊂ Rn _{be an open set and 1 ≤ p < +∞. If U b}

Ω, then the set of all the u ∈ W1,p_{(Ω; R}m_{) such that u}

|U is continuous and

piecewise affine is dense in W1,p_{(Ω; R}m). If Ω has bounded Lipschitz boundary, then the set of continuous and piecewise affine φ ∈ W1,p_{(Ω; R}m_{) is dense in}

W1,p_{(Ω; R}m_).

Moreover we shall need the following Theorems.

Theorem 2.3.3 Let Y be a n-cube in Rn_{, 1 ≤ p ≤ +∞ and let u ∈ L}p

loc (Rn; Rm)

be Y -periodic and v ∈ W1,p_loc_(Rn_{; R}m_{) be such that Dv is Y -periodic. For ε > 0}

define uε(x) := u(x_ε) and vε(x) := εv(x_ε). Then as ε → 0 we have that

uε * 1 |Y | Z Y u(x)dx (* if p = ∞)∗ (2.23) in Lp_{(Ω; R}m_{) for any bounded open set Ω ⊂ R}n and

vε * 1 |Y | Z Y Dv(x)dx (* if p = ∞)∗ (2.24) in W1,p_{(Ω; R}m_{) for any bounded open set Ω ⊂ R}n.

The following theorem can be found in [17].

Theorem 2.3.4 If f : Rm → [0, +∞] is convex and lower semicontinuous, then the functional

F (u) = Z

Ω

f (u(x))dx (2.25)

is sequentially lower semicontinuous on L1_{(Ω; R}m) with respect to the conver-gence in D0_{(Ω; R}m).

The following theorem (which can be found in [18]) allows sometimes to pass from bounded sequences in W1,p_{(Ω; R}m_{) to sequences with equi-integrable p-th}

power of the gradient.

Theorem 2.3.5 If Ω ⊂ Rn _{is a bounded open set and if (u}

k)k⊂ W1,p(Ω; Rm)

is a bounded sequence, there exists a subsequence ujk and a sequence (vk)k ⊂

W1,p_{(Ω; R}m_{) such that |Dv}

k|p is equi-integrabile and

lim

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2.4 Almost periodic functions

Definition 2.4.1 A set T ⊆ Rn is relatively dense if there exists an inclusion lenght, i.e. if there exists L > 0 such that T + [0, L)n_{= R}n

Suppose in the following that (X, || · · · ||) is a complex Banach space.

Definition 2.4.2 τ ∈ Rn _{is said to be an ε-almost period for f : R}n _{→ X if}

sup

x∈Rn

||f (x + τ ) − f (x)|| < +∞ .

f is almost periodic if it is continuous and, for every ε > 0, it has a relatively dense set of ε-almost periods.

For the study of almost periodic functions see [6] and [20]. The following theorem is basic in the theory of almost periodic functions.

Theorem 2.4.3 f : Rn → X is almost periodic if and only if there exists a sequence of trigonometric polinomials Pk(x) =

Prk

j=1a k

j exp (i(λkj, x)) such that

lim

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Chapter 3 Γ-convergence

3.1 From Weierstrass’s theorem to Γ-

conver-gence

We are going to define a particular notion of convergence, namely Γ-convergence that will prove to be very useful. To understand the reason why it has been defined, we shall firstly state the following well known Weierstrass’s theorem and its proof.

In all this chapter (X, d) will be a metric space, usually abbreviated with X. For results on Γ-limits one can look at [14] and at [13].

Definition 3.1.1 A function f : X → R is coercive if , ∀t ∈ R, {f ≤ t} is compact. A function f : X → R is mildly-coercive if there exists a non-empty compact set K ⊆ X such that infXf = infKf . A sequence of functions

fj : X → R is equi-mildly-coercive if there exists a non-empty compact set

K ⊆ X such that , ∀j ∈ N, infXfj = infKfj.

Obviously a coercive lower semicontinuous function is also mildly-coercive. A standard interesting example of a mildly-coercive function that is not coer-cive is given by any periodic lower semicontinuous function f : Rn_{→ R.}

Theorem 3.1.2 If f : X → R is mildly-coercive, then there exists minXf ,

it equals infXf , and the minimum points of f are all the limits of converging

sequences (xi) such that ∃ limjf (xj) = infXf .

Proof : Since f ≤ f we have infXf ≤ infXf and since the constant

function g = inf f is trivially lower semicontinuous and g ≤ f we have g ≤ f and so infXf ≤ infX f and equality holds. Let (xj) be a sequence such that

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CHAPTER 3. Γ-CONVERGENCE

that infXf = infKf , then there exists a sequence (xj) ⊆ K like that), then

we have inf

X f ≤ f (x) ≤ lim infj f (xj) ≤ lim infj f (xj) ≤ infX f = infX f .

We want to define a convergence with the following feature: if (fj) is a

sequence of equi-mildly-coercive functions on X and if (fj) Γ-converges to f

then

∃ min

X f = lim infX fj.

This is clearly and important requirement for a good variational convergence. Following the proof of theorem (3.1.2), we take a sequence (xj) ⊆ K such that

xj → x and

lim inf

j fj(xj) = lim infj infX fj;

then we have inf

X f ≤ f (x) . . . lim infj fj(xj) = lim infj infX fj, (3.1)

so if we could substitue the ellipsis by ’≤’ and if we had lim sup

j

fj(xj) ≤ inf

X f, (3.2)

then all inequalities in (3.1) and (3.2) would become equalities. Thus we are led to the following definition.

Definition 3.1.3 We say that a sequence of functions fj : X → R

Γ(d)-converges to f : X → R if ∀x we have 1. xj → x implies f (x) ≤ lim inf j fj(xj), (3.3) 2. ∃ xj → x such that lim sup j fj(xj) ≤ f (x), (3.4)

or, which is equivalent by (3.3), lim

j fj(xj) = f (x). (3.5)

f is called the Γ(d)- limit of (fj) and we write f = Γ(d)-limjfj. When no

confusion can arise we shall omit the dependence on the metric d. When we are dealing with Lp _{spaces with the usual distance, we write simply}

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3.2. Γ-LIMINF AND Γ-LIMSUP

Remark 3.1.4 We observe that the definition can be given for an arbitrary topological space, so in particular equivalent distances (that is distances which give the same topology) have the same Γ-converging sequences.

We obtain at once the desired theorem

Theorem 3.1.5 If fj : X → R is a sequence of equi-mildly-coercive functions

that Γ(d)-converges to f : X → R then ∃ minXf, ∃ limjinfXfj and

min

X f = limj infX fj. (3.6)

Moreover, given a sequence (xj) such that limjfj(xj) = limjinfX fj, any limit

point x of (xj) is a minimum point for f , and there exist a sequence like that

which admits limit points.

3.2 Γ-liminf and Γ-limsup

As for the usual limits, we can define the Γ- lim inf and the Γ- lim sup, which will always exist, and which are equal iff the Γ- lim exists:

Theorem 3.2.1 Let fj : X → R, x ∈ X, λ ∈ R and N (x) the family of all

neighbourhoods of x . The following statements are equivalent. 1.

xj → x implies λ ≤ lim inf

j fj(xj) and (3.7)

∃xj → x such that λ = lim inf

j fj(xj); (3.8)

2. λ = min{lim infjfj(xj) : xj → x}

3. λ = inf{lim infjfj(xj) : xj → x}

4. λ = sup

U ∈N (x)

lim infj inf y∈Ufj(y).

Theorem 3.2.2 Let fj : X → R, x ∈ X, λ ∈ R and N (x) the family of all

neighbourhoods of x . The following statements are equivalent. 1.

xj → x implies λ ≤ lim sup j

fj(xj) and (3.9)

∃xj → x such that λ = lim sup j

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2. λ = min{lim sup_jfj(xj) : xj → x}

3. λ = inf{lim sup_jfj(xj) : xj → x}

4. λ = sup

U ∈N (x)

lim sup_j inf

y∈Ufj(y).

Definition 3.2.3 We say that λ ∈ R is the Γ(d)- lower limit of the sequence of functions fj : X → R at x and we write

λ = Γ(d)- lim inf

j fj(x) (3.11)

if any of the equivalent conditions of proposition 3.2.1 is satisfied. We say that λ ∈ R is the Γ(d)- upper limit of the sequence of functions fj : X → R at x

and we write

λ = Γ(d)- lim sup

j

fj(x) (3.12)

if any of the equivalent conditions of proposition 3.2.2 is satisfied. If we have Γ(d)- lim inf

j fj(x) = λ = Γ(d)- lim sup_j fj(x) (3.13)

then λ is called the Γ(d)- limit of (fj) at x and we write λ = Γ(d)- limjfj(x).

When no confusion can arise we shall omit the dependence on the metric d.

3.3 Properties of Γ-convergence

Now we state the properties of Γ- convergence that we will need thereafter. Keeping in mind what happens when dealing with limits coming from a topol-ogy, we look for analogous properties for Γ-limits. All the following properties come simply from the definitions.

Remark 3.3.1 If f = Γ- limjfj then the following properties hold:

1. f is lower semicontinuous

2. every subsequence of (fj) Γ-converges to f

3. if g is continuous then g + fj Γ-converges to g + f .

Remark 3.3.2 The following relations between the different kind of conver-gences hold:

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3.3. PROPERTIES OF Γ-CONVERGENCE

2. if fj ↓ f then fj Γ-converges to f

Moreover we have that inequalities and convexity are conserved by Γ- limits and “the sum is not Γ continuous”, since generally f + g 6= f + g; more precisely:

Remark 3.3.3 If (X, d) is a topological vector space then: 1. if every fj is convex then Γ(d)-lim supjfj is convex

2. if fj ≤ gj then Γ(d)-lim infjfj ≤ Γ(d)-lim infjgj and

Γ(d)-lim sup_jfj ≤ Γ(d)-lim supjgj

3. Γ(d)-lim infj(f + g)j ≥ Γ(d)-lim infjfj + Γ(d)-lim infjgj and

Γ(d)-lim sup_j(f + g)j ≥ Γ(d)-lim supjfj + Γ(d)-lim supjgj with possible

inequalities

4. if fj ≥ gj + hj than Γ(d)-lim infjfj ≥ Γ(d)-lim infjgj + Γ(d)-lim infjhj

and

Γ(d)-lim sup_jfj ≥ Γ(d)-lim supjgj + Γ(d)-lim supjhj

5. if α > 0 then Γ(d)-lim infjαfj = αΓ(d)-lim infjfj and Γ(d)-lim supjαfj =

αΓ(d)-lim sup_jfj .

We note that property (3.3.3.(1)) is not true for the Γ(d)-lim inf and properties (3.3.3.(3)) and (3.3.3.(4)) are true whenever the sums are well defined. Now let us see two important properties, that we will use many times. As for every convergence coming from a topology we have

Theorem 3.3.4 fj Γ(d)-converges to f at x iff for every subsequence there

exists a further subsequence that Γ(d)-converges to f at x.

Finally separable metric spaces are sequentially compact with respect to Γ-convergence, in the following sense:

Theorem 3.3.5 If (X, d) is a separable metric space every sequence fj : X →

R admits a subsequence that Γ(d)-converges at all x ∈ X.

Proof Let Uk be a countable base of open sets. Since R is compact there exists

an increasing sequence of integers (σ0

j)j such that ∃ limjinfy∈U0fσ_j0(y). We

define (σk

j)j to be any subsequence of (σjk−1)j such that ∃ limjinfy∈Ukfσk_j(y).

Then, if jk = σkk is the diagonal sequence, we have

∀l ∈ N lim inf

k y∈Uinfl

fjk(y) = lim sup

k

inf

y∈Ul

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so using as definition property 4 of (3.2.1) and of (3.2.2) we have that the Γ-liminf and the Γ-limsup coincide.

Finally we will need the following theorem.

Theorem 3.3.6 Take f, fj : X → [0, +∞), where X is a topological

metriz-able vector space. If (fj) is a sequence of quadratic forms which Γ-converges

to f , then f is a quadratic form.

Proof : f is a quadratic form if and only if it is positively homogeneous of degree 2 (that is f (λA) = |λ|2_{f (A) whenever λ ≥ 0) and f (x + y) +}

f (x − y) = 2f (x) + 2f (y). The Γ-limit of positively homogeneous functions is obviously positively homogeneous of the same degree. Inequality f (x + y) + f (x − y) ≤ 2f (x) + 2f (y) comes directly if we take xj → x and yj → y

such that lim infjfj(xj) = f (x) and lim infjfj(yj) = f (y) and we consider two

subsequences xjk and yjk such that ∃ limkfjk(xjk) and ∃ limkfjk(yjk). Infact

then xj + yj → x + y, xj− yj → x − y and so we can write

f (x + y) + f (x − y) ≤ lim inf j fj(xj + yj) + lim infj fj(xj− yj) ≤ ≤ lim inf k (fjk(xjk+ yjk) + fjk(xjk − yjk)) = = lim inf k (2fjk(xjk) + 2fjk(yjk)) = 2f (x) + 2f (y) .

The opposite inequality follows if we take z = x + y and t = x − y and we write

4f (x) + 4f (y) = f (2x) + f (2y) = f (z + t) + f (z − t) ≤ ≤ 2f (z) + 2f (t) = 2f (x + y) + f (x − y) .

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Chapter 4 Integral representation

4.1 An integral representation theorem

The direct methods of Γ-convergence for integral fuctionals consists in proving the existence of Γ-converging sequences and in recovering enough information on the structure of the Γ-limit as to obtain a suitable representation of the limit. We will consider Γ-limits of fuctionals of the form

F (u, U ) = Z

U

f (x, Du(x))dx, (4.1)

with u ∈ W1,p_{(Ω; R}m) (the Sobolev space) and U ∈ A(Ω) (the family of all open subsets of Ω)) We will use the localization method, which considers at the same time the dependence of the Γ-limits on the function u and on the domain of integration U . We address the problem of proving that an abstract functional F (u, U ) is of the form (4.1). From now Ω, U and V will be open subsets of Rn_{. A collection of integral representation theorems can be found}

in [10].

Theorem 4.1.1 Let 1 ≤ p < ∞. If F : W1,p_{(Ω; R}m_{) × A(Ω) → [0, +∞),}

then there exists a quasiconvex function f : Ω × Rm×n _{→ [0, ∞) satisfying the}

growth condition

0 ≤ f (x, A) ≤ c(a(x) + |A|p) _{for a.e. x ∈ Ω and ∀A ∈ R}m×n (4.2) and such that

F (u, U ) = Z

U

f (x, Du(x))dx ∀u ∈ W1,p

(Ω; Rm) and ∀U ∈ A(Ω) (4.3) if and only if the following conditions hold:

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CHAPTER 4. INTEGRAL REPRESENTATION

1. (measure property) ∀u the function F (u, ·) is the restriction of a Borel measure to A(Ω)

2. (growth condition) there exists c > 0 and a ∈ L1(Ω) such that F (u, U ) ≤ c

Z

U

(a(x)+|Du(x)|p)dx ∀u ∈ W1,p

(Ω; Rm) and ∀U ∈ A(Ω) 3. (translation invariance in u) F (u, U ) = F (u + z, U ) _{∀z ∈ R}n_{, ∀U ∈}

A(Ω) and ∀u ∈ W1,p

(Ω; Rm)

4. ∀U ∈ A(Ω) the function F (·, U ) is sequentially lower semicontinuous with respect to the weak convergence.

If there exists such a function f , then it is unique up to equality x a.e. : g is another such function iff g(x, A) = f (x, A) for all A and for almost every x. Moreover f (x, A) is indipendent from x iff F is “translation invariant in U ”, or more precisely iff

5. F (Ax, B(y, ρ)) = F (Ax, B(z, ρ))

for all A ∈ Rm×n, y, z ∈ Ω, ρ > 0 such that B(y, ρ) ∪ B(z, ρ) ⊆ Ω . In this case there exists h : Rm×n _{→ [0, ∞) quasiconvex and such that}

0 ≤ h(A) ≤ c(1 + |A|p₎ _{∀A ∈ R}m×n

F (u, U ) =R_Uh(Du(x))dx ∀u ∈ W1,p

(Ω; Rm) and ∀U ∈ A(Ω) (4.4) Proof One implication is obvious, the other one goes as follows:

Step 1: Representation on piecewise affine functions.

Fix A ∈ Rm×n_{, then F (Ax, ·) can be extended to a Borel measure which, by}

condition (2), is absolutely continuous with respect to the Lebegue measure. So by the Radon-Nikodym theorem there exists a function f (·, A) ∈ L1(Ω) such that

F (Ax, U ) = Z

U

f (x, A)dx ∀U ∈ A(Ω),

and by a Lebegue theorem, if B(y, ρ) = {x ∈ Rn _{:k x − y k< ρ}, we can choose}

for f (·, A) the following version f (y, A) = lim sup

ρ→0+

F (Ax, B(y, ρ))

|B(y, ρ)| . (4.5)

From the same theorem, if g is a measurable function that satisfies (4.2) and (4.3), then g(x, A) = f (x, A) ∀A and for a.e. x, so that function is unique

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4.1. AN INTEGRAL REPRESENTATION THEOREM

up to equality x a.e. . Then by condition (2) we have (4.2). Now let u be piecewise affine on U ∈ A(Ω), such that we can write

u|U = N

X

j=1

χUj(Ajx + zj),

with Uj disjoint open subsets of U with |U \

SN j=1Uj| = 0 , Aj ∈ Rm×n and zj ∈ Rm. Then we have F (u, U ) =PN j=1F (u, Uj) = PN j=1F (Ajx + zj, Uj) = PN j=1F (Ajx, Uj) = =PN j=1 R Ujf (x, Aj)dx = PN j=1 R Ujf (x, Du)dx = R Uf (x, Du)dx,

so representation (4.3) holds for such functions. Step 2: Continuity of f

We want to show that f is continuous, so that by theorem (2.1.4) and remark (2.1.6) it is quasiconvex. It is sufficient to show that f (y, ·) rank one convex, because then by remark (2.1.11) it is locally Lipschitz. To show that f given by (4.5) is rank-one convex it is sufficient to show that, fixed t ∈ (0, 1) and A, B ∈ Rm×n _{such that B − A = a ⊗ b, then}

F ((tB + (1 − t)A)x, B(y, ρ)) ≤ tF ((Bx, B(y, ρ)) + (1 − t)F (Ax, B(y, ρ)). Let us define v(x) = ( Ax + (b, x)a − (1 − t)ja _{j ∈ Z, j ≤ (b, x) < j + t} Ax + (1 + j)ta _{j ∈ Z, j + t ≤ (b, x) < j + 1} EA = {x ∈ Rn: ∃j ∈ Z : j + t ≤ (b, x) < j + 1} EB = {x ∈ Rn: ∃j ∈ Z : j ≤ (b, x) < j + t}

then v ∈ W_loc1,∞_(Rn_{; R}m_{) and, if we set u}

j(x) = 1_jv(jx), we have Duj = A on 1

jEA and Duj = B on 1

jEB and by theorem (2.3.3) that for every bounded

open set V uj ∗ * (tB + (1 − t)A)x weakly∗ in W1,∞_{(V ; R}m₎ χ1 jEA ∗ * 1 − t weakly∗ in L∞(V ) χ1 jEB ∗ * t weakly∗ in L∞(V ).

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So we have

F ((tB + (1 − t)A)x, B(y, ρ)) ≤ lim infjF (uj, B(y, ρ)) =

= lim infj

F (Ax,1_jEA∩ B(y, ρ) + F (Bx,1_jEA∩ A(y, ρ))

= lim infj R B(y,ρ)χ1jEAf (x, A)dx + R B(y,ρ)χ1jEBf (x, B)dx = (1 − t)R B(y,ρ)f (x, A)dx + t R B(y,ρ)f (x, B)dx =

= tF ((Bx, B(y, ρ)) + (1 − t)F (Ax, B(y, ρ)). Step 3: Extension of the representation by density

By remark (2.1.1) the functional u 7→R_Uf (x, Du)dx is continuous with respect to the strong convergence in W1,p_{(Ω; R}m_{). Take U b Ω, so, given any u, there} exists a sequence uj converging strongly to u in W1,p(Ω; Rm) and such that

uj|U is piecewise affine as before. Then

F (u, U ) ≤ lim infjF (uj, U ) = lim infj

R

Uf (x, Duj)dx =

R

Uf (x, Du)dx.

Now fix u and define G(v, U ) := F (u + v, U ), then G satisfies the hypotheses of this theorem (with different c and a(x) in (2)) and so, by what preceeds, there exists a quasiconvex function ψ such that

G(v, U ) ≤ Z

U

ψ(x, Dv)dx.

for all U b Ω and v ∈ W1,p_{(Ω; R}m_{), with equality if v is piecewise affine on U .}

So if U b Ω and uj converges strongly to u in W1,p(Ω; Rm) and is such that

uj|U is piecewise affine, we have

R Uψ(x, 0)dx = G(0, U ) = F (u, U ) ≤ R Uf (x, Du)dx = = limj R

Uf (x, Duj)dx = limjF (uj, U ) = limjG(uj − u, U ) ≤

≤ limj R Uψ(x, Duj − Du)dx = R Uψ(x, 0)dx , so clearly F (u, U ) = Z U f (x, Du)dx

holds whenever U b Ω and so, for the criterion of coincidence of measures, for all U ⊆ Ω.

Step 4 : Translation invariance in U f (y, A) = lim sup

ρ→0+

F (Ax, B(y, ρ))

|B(y, ρ)| = lim supρ→0+

F (Ax, B(z, ρ))

|B(z, ρ)| = f (z, A) and we can take ||a||L1 instead of a(x).

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4.2. INCREASING SET FUNCTIONS

4.2 Increasing set functions

Now we address the task of characterizing the functions which are restriction of a Borel measure to A(Ω). We give first some definitions.

Definition 4.2.1 A set function α : A(Ω) → [0, +∞] is called an increasing set function if α(∅) = 0 and if it is monotone increasing, that is if V ⊆ U implies α(V ) ≤ α(U ). It is subadditive if

α(U ∪ V ) ≤ α(U ) + α(V ) (4.6)

for all U and V, it is superadditive if

α(U ∪ V ) ≥ α(U ) + α(V ) (4.7)

whenever U ∩ V = ∅, and it is inner regular if

α(U ) = sup{α(V ) : V b U } (4.8) for all U and V.

The proof of the following theorem can be found in [15].

Theorem 4.2.2 (Measure property criterion) Let α : A(Ω) → [0, +∞] be an increasing set function. Then the following properties are equivalent.

1. α is the restriction to A(Ω) of a Borel measure on Ω 2. α is subadditive, superadditive and inner regular 3. The set function

β(E) = inf{α(U ) : U ∈ A(Ω), E ⊆ U } (4.9) is a Borel measure on Ω.

4.3 The fundamental estimate

Willing to use the theorem (4.1.1) to prove that the Γ-limits that we will consider admit an integral representation, having in mind the measure property criterion, we are looking for properties that ensure that the Γ-limits define increasing set functions which are subadditive, superadditive and inner regular. We need the following basic technical ingredient, the so-called “fundamental estimate”, which permits to “join fuctions without introducing a large error”. Dealing with that estimate, unless otherwise stated, we suppose 1 ≤ p < ∞. The following version of the Lp_{-fundamental estimate is due to [8], and is a}

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Definition 4.3.1 Given U, U0 ∈ A(Ω) such that U0

b U , we say that φ is a cut-off fuction between U0 and U if φ ∈ C_c∞(U ), 0 ≤ φ ≤ 1 and φ|U0 = 1.

Definition 4.3.2 F : Lp_{(Ω; R}n_{) × A(Ω) → [0, +∞] is said to satisfy the L}p

-fundamental estimate if whenever U, U0, V ∈ A(Ω), U0 _{b U and σ > 0, there} exist M > 0 such that ∀u, v ∈ Lp_{(Ω; R}n_{) there exists a cut-off fuction φ between}

U0 and U such that

F (φu + (1 − φ)v, U0∪ V ) ≤ (1 + σ)(F (u, U ) + F (v, V )) + +σ + M

Z

(U ∩V )\U0

|u − v|p_dx _{. (4.10)}

As usual, when working with integral functionals, we need to suppose growth and coercivity conditions to be satisfied. We will be interested in two cases, both of which satisfy the fundamental estimate.

Definition 4.3.3 If f : Ω × Rm _{× R}m×n _{→ [0, +∞) is a Borel function, we}

say that it satisfies the standard growth conditions of order p if there exist α, β > 0 such that

α|A|p ≤ f (x, s, A) ≤ β(1 + |A|p₎ _(4.11)

for all x ∈ Ω, s ∈ Rm_{, A ∈ R}m×n_.

We say that F : W1,p_{(Ω; R}m_{) × A(Ω) → [0, ∞) belongs to the class F (α, β, p)}

iff there exists a Borel function f : Ω × Rm×Rm×n _{→ [0, +∞) satisfying (4.11)}

and such that

F (u, U ) = Z

U

f (x, u, Du(x))dx (4.12)

for all u ∈ W1,p_{(Ω; R}m_{), U ∈ A(Ω).}

In all the following, when we have a functional defined for u ∈ W1,p_{(Ω; R}m_),

we shall consider that functional as defined on all Lp_{(Ω; R}m_{), meaning that}

it is extended to +∞ on Lp_{(Ω; R}m) \ W1,p_{(Ω; R}m) (we take this extension to mantain the property of being lower semicontinuous ).

For the following theorem we need to suppose that Ω is bounded, so all the results we will obtain in homogenization theory will be about bounded sets. Theorem 4.3.4 The family F (α, β, p) satisfies the Lp_{-fundamental estimate}

uniformly (which means that every F ∈ F (α, β, p) satisfies the fundamental estimate and the constant M can be chosen uniformly on F (α, β, p)).

Proof : Fix σ > 0 and U0, U, V ∈ A(Ω) with U0 _{b U . From now on, if} A, B ⊂ Rn_{, we will write d(A, B) instead of inf{||x − y|| : x ∈ A, y ∈ B} ( and}

if A = {x}, simply d(x, B)), and we define Ab

a = {x ∈ A : a < d(x, U

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4.3. THE FUNDAMENTAL ESTIMATE

Ab_a = {x ∈ A : a ≤ d(x, U0) ≤ b} and we give analogous definitions for Aa, Ab, A a, A

b_{. Taken δ = d(U}0_{, ∂U ), η and r such that 0 < η < η + r < δ,}

there exists a cut off function φ between Ur and Uη+r with ||Dφ|| ≤ 2/η.

Chose u, v ∈ W1,p_{(Ω, R}m_{) (otherwise the inequality is obvious), then if w =}

uφ + (1 − φ)v we have F (w, U0∪ V ) = F (u, (U0∪ V )r) + F (v, (U0∪ V ) η+r) + F (w, (U 0_{∪ V )}η+r η ) ≤ ≤ F (u, U ) + F (v, V ) + β Z Vηη+r 1 + |φDu + (1 − φ)Dv + (u − v)Dφ|pdx ≤ ≤ F (u, U ) + F (v, V ) + β2p−1₂p−1 Z Vηη+r 1 + |Du|p+ |Dv|p+ 2 p ηp|u − v| p_dx

Now define the measure µ(E) := β4p−1R

E1 + |Du| p _{+ |Dv|}p_{dx and take} N ∈ N \ {0}, then N X k=1 µ(V_δ(k−1)/Nδk/N ) ≤ µ(U ∩ V ) ≤ β4p−1 |U ∩ V | + F (u, U ) + F (v, V ) α . So for every such N there exists k ∈ {1, . . . , N } such that

Z V_δ(k−1)/Nδk/N 1 + |Du|p+ |Dv|pdx ≤ |U ∩ V | N + F (u, U ) + F (v, V ) N α so taken η = δ/N, r = δ(k − 1)/N and N ≥ max{β4 p−1_{|U ∩ V |, β4}p−1_/α} σ M = β23p−2_Np δp

we have the thesis.

We have the same conclusion with the following less natural hypotesis: Let g : Ω × Mm×n → [0, +∞) be a Borel function, convex in the second variable and such that there exists C > 0 such that for all x ∈ Ω, A ∈ Rm×n

g(x, A) ≤ C(1 + |A|p), g(x, 2A) ≤ C(1 + g(x, A)). (4.13) Definition 4.3.5 We say that F : W1,1_{(Ω; R}m_{) × A(Ω) → [0, ∞) belongs to}

the class F (g) iff there exists a Borel function f : Ω × Rm× Rm×n_{→ [0, +∞)}

such that

g(x, A) ≤ f (x, s, A) ≤ c(1 + g(x, A)) for every x ∈ Ω, s ∈ Rm, A ∈ Rm×n and

F (u, U ) = Z

U

f (x, u, Du(x))dx for all u ∈ W1,1_{(Ω; R}m_{), U ∈ A(Ω).}

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CHAPTER 4. INTEGRAL REPRESENTATION If F ∈ F (g) for ε > 0 define Fε(u, U ) = Z U fx ε, u ε, Du(x) dx , (4.14)

then Fε(g) is the set of all Fε such that F ∈ F (g) and ε > 0. We consider F

and Fε to be defined on Lp(Ω; Rm) by taking the restriction to Lp(Ω; Rm) ∩

W1,1_{(Ω; R}m_{) and the +∞ extension on L}p_{(Ω; R}m_{) \ W}1,1_{(Ω; R}m_).

Theorem 4.3.6 If Ω is bounded the family Fε(g) satisfies the Lp-fundamental

estimate uniformly.

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Chapter 5 Γ-limits of functionals

5.1 Regularity of Γ-limits

From the fundamental estimate we will derive some inequalities for Γ-limits, which will permit us to proof that Γ-limits are subaddittive and inner regular increasing set functions. Now we need to define the concept of satisfing the Lp

-fundamental estimate as ε → 0, a slightly weaker condition than the uniform one:

Definition 5.1.1 If Fε : Lp(Ω; Rn) × A(Ω) → [0, +∞], we say that (Fε)ε>0

satisfies the fundamental estimate as ε → 0 if whenever U, U0, V ∈ A(Ω), U0 _{b U and σ > 0, there exist M, ε}0 > 0 such that ∀ε < ε0, u, v ∈ Lp_{(Ω; R}n₎

there exists a cut-off fuction φ between U0 and U such that

Fε(φu + (1 − φ)v, U0∪ V ) ≤ (1 + σ)(Fε(u, U ) + Fε(v, V )) +

+σ + M Z

(U ∩V )\U0

|u − v|pdx (5.1) Obviously an analogous definition holds for (Fj)j∈N with j → +∞.

From now on we will write simply Γ(Lp)- limjFj(u, U ), meaning the Γ(Lp

)-limit of the functionals u 7→ Fj(u, U ) (and analogously for the Γ-liminf and

the Γ-limsup). References for this chapter are [13] and [9].

Theorem 5.1.2 Let p ∈ [1, +∞) and Fε : Lp(Ω; Rm) × A(Ω) → [0, +∞] and

(Fε)ε>0 be a family of functionals, satisfying the Lp-fundamental estimate as

ε → 0. Given a sequence (εj) of positive real numbers converging to zero define

F0(u, U ) := Γ(Lp)- lim inf

j Fεj(u, U )

F00(u, U ) := Γ(Lp)- lim sup

j

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CHAPTER 5. Γ-LIMITS OF FUNCTIONALS

then we have

F0(u, U0∪ V ) ≤ F0(u, U ) + F00(u, V ) (5.2) F00(u, U0 ∪ V ) ≤ F00(u, U ) + F00(u, V ) (5.3) for all u ∈ Lp_{(Ω, R}m_{) and U}0_{, U, V ∈ A(Ω) with U}0

b U .

Proof : Let us prove (5.2), the proof of (5.3) being completely analogous. Take uj converging to u in Lp(U, Rm) and vj converging to u in Lp(V, Rm)

such that

F0(u, U ) = lim inf

j Fεj(uj, U ) and F 00

(u, V ) = lim sup

j

Fεj(vj, U ) .

From now on consider, instead of uj, with the same notation, the extensions

on the whole Ω which value on Ω \ U is u, and analogously for vj take the

extension on the whole Ω which value on Ω \ V is u. Then uj and vj converge

strongly in Lp_{(Ω; R}m_{) to u. Fix σ > 0 and U}0_{, U, V ∈ A(Ω) with U}0

b U , then apply the Lp estimate to the functions uj, vj finding M, ε0 such that for

all ε < ε0, there exists cutoff functions φj between U0 and U such that, if we

define wj := φuj+ (1 − φ)vj, we have

Fεj(wj, U 0_{∪ V ) ≤ (1 + σ)(F} εj(uj, U ) + Fεj(vj, V )) + +σ + M Z (U ∩V )\U0 |uj − vj|pdx (5.4)

Since ||uj− vj||Lp_(Ω;Rn₎ → 0 and ||φ||p

L∞ ≤ 1 we have w_j → u in Lp(Ω; Rm) and

F0(u, U0∪ V ) ≤ lim inf

j Fεj(wj, U

0_{∪ V ) ≤}

≤ (1 + σ)(lim inf

j Fεj(uj, U ) + lim sup_j Fεj(vj, V )) + σ =

= (1 + σ)(F0(u, U ) + F00(u, V )) + σ and by the arbitrariness of σ we have (5.2).

Theorem 5.1.3 Taken q ∈ [1, +∞), under the same hypothesis of Theo-rem (5.1.2) if we further suppose that Ω is bounded and that for all u ∈ W1, q_{(Ω; R}m_{) ∩ L}p_{(Ω; R}m_{) the limits F}0_{(u, ·) and F}00_{(u, ·) are increasing set}

functions and

F00(u, U ) ≤ c Z

U

(1 + |Du|p)dx ∀U ∈ A(Ω) (5.5) then F0(u, ·) and F00(u, ·) are inner regular increasing set functions and F00(u, ·) is subadditive.

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5.2. EXISTENCE OF Γ-LIMITS

Proof : Now take K compact contained in W ∈ A(Ω), choose U0, U ∈ A(Ω) such that K ⊂ U0 _{b U b W and define V = W \ K. By (5.2) we have}

F0(u, W ) ≤ F0(u, U0∪ V ) ≤ F0(u, U ) + F00(u, W \ K) F00(u, W ) ≤ F00(u, U0 ∪ V ) ≤ F00(u, U ) + F00(u, W \ K) so taken u ∈ W1, q_{(Ω; R}m) ∩ Lp_{(Ω; R}m) by (5.5) we have

F0(u, W ) ≤ sup{F0_{(u, U ) : U b W } + c} Z

W \K

(1 + |Du|p)dx F00(u, W ) ≤ sup{F00_{(u, U ) : U b W } + c}

Z

W \K

(1 + |Du|p)dx

so letting |W \ K| → 0, the increasing set functions satisfy the inequalities F0(u, W ) ≤ sup{F0_{(u, U ) : U b W }}

F00(u, W ) ≤ sup{F00_{(u, U ) : U b W }}

and so the equalities hold and they are inner regular. Now fix U, V ∈ A(Ω) and take a sequence of compact sets Knsuch that Kn⊆ Kn+1◦ and U =

S

nKn.

Then by (5.3) with U0 = Kn we have

F00(u, K_n◦∪ V ) ≤ F00(u, U ) + F00(u, V ) and taking the sup_n by the inner regularity we have

F00(u, U ∪ V ) ≤ F00(u, U ) + F00(u, V ).

5.2 Existence of Γ-limits

We are going to show a stronger Γ-compactness result, which takes into account simultaneously the dependence of the functional on the functions and on the sets. The proof in the vector valued case is given by [19].

Theorem 5.2.1 Take 1 ≤ p ≤ +∞ and let Fj : Lp(Ω; Rm) × A(Ω) → [0, +∞]

be a sequence of functionals. Define

F0(u, U ) := Γ(Lp)- lim inf

j Fj(u, U )

F00(u, U ) := Γ(Lp)- lim sup

j

Fj(u, U )

and suppose that, for every u ∈ A ⊂ Lp_{(Ω; R}m_{), they are inner regular}

in-creasing set functions. Then there exists a subsequence (jk)k such that, for

all U ∈ A(Ω), the sequence of functionals Fjk(u, U ) Γ(L

p_{)-converges at all}

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We will always apply theorem (5.2.1) taking A = W1,p_{(Ω; R}m₎

Proof : Let R = (Rj)j∈N be the family of all finite unions of open balls in Ω

with rational vertices and radii. By proposition (3.3.5) there is a subsequence (σ0

j)j such that ∃ Γ(Lp)- limjFσ0

j(u, Rj). We define inductively (σ

k

j)j to be any

subsequence of (σ_jk−1)j such that ∃ Γ(Lp)- limjFσk

j(u, Rk). Then, if jk = σ

k k is

the diagonal sequence, we have that

∀R ∈ R ∃Γ(Lp_{)- lim}

k Fjk(u, R)

Moreover F0(R) = F00(R) _{∀R ∈ R and ∀ V, U ∈ A(Ω) such that V b U there} exists R ∈ R such that V b R b U , so we have

Γ(Lp_{)- lim inf}

kFjk(u, U ) = F

0_{(U ) = sup{F}0

(V ) : V ∈ A(Ω), V b U } = = sup{F0_{(R) : R ∈ R, R b U } = sup{F}00_{(R) : R ∈ R, R b U } =} = sup{F00_{(V ) : V ∈ A(Ω), V b U } = F}00(U ) = Γ(Lp_{)- lim sup}

kFjk(u, U )

Observing that Γ-limits of functionals of the form F (u, U ) =R

Uf (x, Du(x))dx

with positive f are increasing set functions by remark (3.3.3.(2)), and that they are super-additive by (3.3.3.(4)), combining Theorem (5.2.1), Theorem (4.2.2), and Theorem (5.1.3) we finally obtain

Theorem 5.2.2 Let Ω be bounded and (Fε)ε>0 be a family in F (α, β, p) or

in Fε(g). Then for every sequence (εj) of positive real numbers converging to

zero there exists a subsequence (εjk) such that for all U ∈ A(Ω) and at all

u ∈ W1,p_{(Ω; R}m)

∃ Γ(Lp_{)- lim}

k Fεjk(u, U ) =: F (u, U )

and F (u, ·) is the restriction of a Borel measure to A(Ω).

5.3 Representation of Γ-limits

In the case of integral functionals F ∈ F (α, β, p) and F ∈ Fε(g) we are now

able to apply the representation theorem to functionals with integrands which “do not depend on the variable u” to obtain an explicit integral representation of the Γ-limit.

Theorem 5.3.1 Let Ω be bounded and take α, β > 0, p ∈ [1, +∞). Let (fε)ε>0

be a family of Borel functions with f : Ω × Rm×n _{→ [0, +∞) satisfying the}

estimate

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5.4. BOUNDARY VALUES AND Γ-LIMITS

for all x ∈ Ω, A ∈ Rm×n_{, ε > 0 and}

Fε(u, U ) :=

Z

U

fε(x, Du(x))dx (5.7)

if U ∈ A(Ω) and u ∈ W1,p_{(Ω; R}m). Then for every sequence (εj) of positive real

numbers converging to zero there exists a subsequence (εjk) and a quasiconvex

function φ : Ω × Rm×n_{→ [0, +∞) satisfying (5.6) such that}

∃ Γ(Lp_{)- lim}

k Fεjk(u, U ) =

Z

U

φ(x, Du)dx (5.8)

for all U ∈ A(Ω) and u ∈ W1,p_{(Ω; R}m_).

Obviously we can obtain the same if f is as in definition (4.3.5) instead of as in definition (4.3.3).

Proof : The theorem (5.2.2) gives the existence of the Γ-limit, that we will call F (u, U ). To apply theorem (4.1.1) we need only to prove property (4), because property (1) has been proven in theorem (5.2.2) and the other properties are obvious. Take uj * u in W1,p(Ω; Rm) and V b Ω with Lipschitz boundary.

Then, for theorem (2.3.1) applied to a minimizing subsequence, there exists a further subsequence ujk such that ujk → u in L

p

(V ; Rm) and so, for remark (3.3.1.(1)), we have

F (u, V ) ≤ lim inf

k F (ujk, V ) = limk F (ujk, V ) = lim infj F (uj, V ) .

So F (·, V ) is lower semicontinuous for every such V . Now take U ∈ A(Ω), for the inner regularity of F (u, ·) we have

F (u, U ) = sup{F (u, V ) : V b U and with Lipschitz boundary } and the sup of lower semicontinuous functions is lower semicontinuous. We stress that in the previous theorem the hypothesis of indipendence of fε

from u was used only to assure that the Γ-limit is translational invariant in u. In section (6.3) we will consider a more general case.

5.4 Boundary values and Γ-limits

If we suppose that the estimate (5.5) is satisfied uniformly, we obtain that the minimizing sequences for the Γ-limits can be taken with the same boundary value as their limits:

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Theorem 5.4.1 Let Fε : W1,p(Ω; Rm) × A(Ω) → [0, +∞] , suppose (Fε)

sat-isfies the Lp_{-fundamental estimate as ε → 0 and that Ω is bounded. Suppose}

that εj → 0+ and

Fεj(v, U ) ≤ c

Z

U

(1 + |Dv|p)dx (5.9)

for all U ∈ A(Ω) and v ∈ W1,p_{(Ω; R}m). Then if φ ∈ W1,p_{(Ω; R}m) and Gφ_ε_j(v) = ( Fεj(v, Ω) if v ∈ φ + W 1,p 0 (Ω; Rm) +∞ otherwise we have Γ(Lp)- lim inf j Fεj(v, Ω) = Γ(L p_{)- lim inf} j G φ εj(v) (5.10) Γ(Lp)- lim sup j Fεj(v, Ω) = Γ(L p_{)- lim sup} j Gφ_ε j(v) (5.11) for all v ∈ φ + W₀1,p_{(Ω; R}m₎

Proof : We prove only (5.10) because (5.11) is completely analogous. One inequality is trivial, so let us show the opposite inequality. Take σ > 0 and uj ∈

W1,p_{(Ω; R}m_{) converging to v in L}p_{(Ω; R}m_{) such that Γ(L}p_{)- lim inf}

jFεj(v, Ω) =

lim infjFεj(uj, Ω). Take K compact and U, U

0 _{∈ A(Ω) with K ⊂ U}0

b U b Ω and define V := Ω \ K. Applying the Lp _{fundamental estimate as ε → 0 with}

u = uj, we find M and ε0 such that for all ε < ε0 there exist φj cutoff functions

between U0 and U such that, once defined wj := φjuj + (1 − φj)v, we have

Fεj(wj, Ω) ≤ (1 + σ)(Fεj(uj, Ω) + Fεj(v, Ω \ K)) + +M ||u − v||p_Lp+ σ ≤ ≤ (1 + σ)(Fεj(uj, Ω) + c Z Ω\K (1 + |Dv|p)dx) + +M ||u − v||p_Lp+ σ.

So taking the lim infj and choosing K such that (1+σ)c

R Ω\K(1+|Dv| p_{)dx) ≤ σ} we obtain Γ(Lp)- lim inf j G φ εj(v) ≤ 2σ + (1 + σ)Γ(L p_{)- lim inf} j Fεj(v, Ω),

so letting σ → 0 we conclude the proof, because wj = v on Ω \ U with U b Ω,

so by v ∈ φ + W₀1,p_{(Ω; R}m) it follows that wj ∈ φ + W 1,p

0 (Ω; Rm).

An analogously the minimizing sequences for the Γ-limits at u periodic can be taken to be periodic (of the same period), with the same proof:

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5.5. Γ-LIMITS OF HOMOGENEOUS FUNCTIONALS

Theorem 5.4.2 Under the hypotheses of theorem (5.4.1) and if Ω = (0, c)n

and Hεj(u) = ( Fεj(u|Ω, Ω) if u ∈ W 1,p # ((0, c) n_{; R}m₎ +∞ otherwise then we have Γ(Lp)- lim inf j Fεj(u, Ω) = Γ(L p_{)- lim inf} j Hεj(u) (5.12) Γ(Lp)- lim sup j Fεj(u, Ω) = Γ(L p )- lim sup j Hεj(u) (5.13) for all u ∈ W_#1,p_{(Ω; R}m_{) .}

5.5 Γ-limits of homogeneous functionals

Let us see a result that will be used to prove the Homogenization closure theorem, which states that when the integrands do not depend on the space variable x the Γ-convergence reduces to pointwise convergence:

Theorem 5.5.1 Take α, β > 0, p ∈ (1, +∞) and let Ω ⊆ Rn _{be an open set.}

Let (εj) be a sequence of positive real numbers converging to zero and (fε)ε>0

be a family of continuous functions with fε : Rm×n → [0, +∞) satisfying the

estimate

α|A|p ≤ fε(A) ≤ β(1 + |A|p) (5.14)

for all A ∈ Rm×n, ε > 0 and let Fε(u, U ) :=

Z

U

fε(Du(x))dx. (5.15)

for any open set U ⊆ Ω and for all u ∈ W1,p_{(U ; R}m_).

Then Qfεj(x) → f (x) for all x ∈ Ω if and only if for any open ball B b Ω and

for all u ∈ W1,p_{(B; R}m_{) we have}

∃ Γ(Lp_{)- lim}

j Fεj(u, B) =

Z

B

f (Du(x))dx . (5.16) In this case f is a quasiconvex function which satisfies the standard growth esti-mate of order p and for any bounded open set U ⊆ Ω and any u ∈ W1,p_{(U ; R}m) we have ∃ Γ(Lp)- lim j Fεj(u, U ) = Z U f (Du(x))dx . (5.17) .

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Proof : Applying theorem (5.3.1) and condition 5 of theorem (4.1.1) we obtain that, for every bounded open set U ⊆ Ω, for every subsequence there exists a further subsequence (εjk)k such that,

∃ Γ(Lp_{)- lim}

k Fεjk(u, V ) =

Z

V

φ(Du(x))dx (5.18)

at all u ∈ W1,p_{(U ; R}m_{) and for any V ∈ A(U ), with φ quasiconvex satisfying}

the standard growth estimate of order p.

Now suppose Qfεj(x) → f (x) for all x; we are going to show that φ = f , so we

obtain that f is quasiconvex and every subsequence of εj admits a subsequence

along which there exists the Γ-limit and it equals R_V f (Du(x))dx. Thus, by (3.3.4), we have one implication and that equality (5.17) holds.

Chose a A ∈ Rm×n _{and an open ball B b U , then by remark (2.1.6) we have}

|B|φ(A) = inf{ Z B φ(Du(x))dx : u ∈ Ax + W₀1,p_{(U ; R}m)} (5.19) |B|Qfε_jk(A) = inf{ Z B fε_jk(Du(x))dx : u ∈ Ax + W 1,p 0 (U ; Rm)} ;(5.20)

by theorem (3.1.5) and (5.4.1) we have ∃ lim k inf{ Z B fε_jk(Du(x))dx : u ∈ Ax + W 1,p 0 (U ; Rm)} = inf{ Z B φ(Du(x))dx : u ∈ Ax + W₀1,p_{(U ; R}m)} (5.21) so with equations (5.19) and (5.20) we obtain |B|φ(A) = limk|B|Qfε_jk(A) =

|B|f (A).

The other implication is similar: for every bounded open set U ⊆ Ω, for every subsequence there exists a further subsequence (εjk)k such that (5.18), so the

hypothesis implies R_Bf (Du(x))dx = R_Bφ(Du(x))dx and so f = φ; so f is quasiconvex. Again by theorem (3.1.5) and (5.4.1) equations (5.21), (5.19), (5.20) hold, so we obtain ∃ limk|B|Qfε_jk(A) = |B|f (A) and the thesis.

So taking fε= f and using remark (3.3.2.(1)) we immediately obtain:

Corollary 5.5.2 Let Ω ⊂ Rn _{be a bounded open set and f : R}m×n_{→ [0, +∞)}

be a continuous function satisfying the estimate

α|A|p ≤ f (A) ≤ β(1 + |A|p₎ _(5.22)

for all A ∈ Rm×n and with α, β > 0, p ∈ (1, +∞). Let F (u) :=

Z

Ω

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5.5. Γ-LIMITS OF HOMOGENEOUS FUNCTIONALS

for all u ∈ W1,p_{(Ω; R}m_{). If F is the lower semicontinuous envelope with respect}

to the Lp_{-topology of the functional F , then}

F (u) = Z

Ω

Qf (Du(x))dx for all u ∈ W1,p_{(Ω; R}m_).

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Chapter 6 Periodic and Almost Periodic

Homogenization

6.1 Periodic homogenization

We shall see that, under periodicity hypothesis, not only some sequences of functionals Fεj(u, U ) Γ-converge to a functional of the form F (u, U ) =

R

Uf (Du(x))dx, but the whole family do that. Our main reference for this

chapter is [9]. The definition of Γ-convergence for families (Fε) is completely

analogous to the definition for sequences (3.1.3), and it is equivalent to saying that for every sequence (εj) of positive real numbers converging to zero we

have that the sequence (Fεj) Γ-converge to a limit which is indipendent from

the sequence (εj). By theorem (3.3.5) we have that the whole family converges

to λ at u if and only if every sequence admits a subsequence along which the family converges to λ at u. So, applying theorem (5.3.1), to show that the whole family converges we need to show that the function φ(x, Du) is indipen-dent from x, from the sequence (εj) and from the bounded open set Ω.

In what follows we take

Fε(u, Ω) =

Z

Ω

f (x

ε, Du(x))dx (6.1)

whenever Ω ⊂ Rn is a bounded open set, and we suppose p ∈ [1, +∞) and that f : Rn_{× R}m×n_{→ [0, +∞) is a Borel function which satisfies the standard}

growth condition of order p and that

f (·, A) is 1-periodic for all A ∈ Rm×n, (6.2) i.e. that f (x + ei, A) = f (x, A) for all x ∈ Rn, A ∈ Rm×n, i ∈ {1, . . . , n}.

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6.1. PERIODIC HOMOGENIZATION

Γ-limit because, due to the standard growth condition, there exists the Γ-limit on Lp _{\ W}1,p _{and it equals +∞: if u is a points where the Γ-lim inf is finite,}

minimizing sequences uj which converge in Lp to u have bounded derivative,

so they admit a weakly convergent subsequence and so u is a Sobolev function.

Theorem 6.1.1 In the previous hypotheses there exists a (unique) quasicon-vex function fhom: Rm×n → [0, +∞) which satisfies the standard growth

condi-tion of order p and such that for every bounded open set Ω ⊂ Rn _{and at every}

u ∈ W1,p_{(Ω; R}m) we have ∃Γ(Lp_{)- lim} ε→0Fε(u, Ω) = Z Ω fhom(Du(x))dx . Moreover ∃ lim t→+∞ 1 tninf Z (0,t)n f (x, A + Du(x))dx : u ∈ W₀1,p((0, t)n_{; R}m) (6.3)

and it equals fhom(A).

Proof : The proof of the existence of limit (6.3) is obtained by taking, with fixed t, a function utapprossimating the inf at t, and finding an approssimation

us of the inf at s t by patching together copies of the ut, defined on the

cubes contained in (0, s)n_{and of the type i + (0, t)}n_{with i ∈ Z}N _{(we stress that}

the only hypothesis used here are the periodicity and the fact that for every A the quantity sup{f (x, A) : x ∈ Rn_{} is finite; in particular the coercivity is not}

used). We will write down explicitely the proof only in the more complicated case of almost periodic functions. By the representation theorem, to prove that the function φ is indipendent from x, we have to show that

Γ(Lp)- lim

k Fεjk(Ax, Bρ(y)) = Γ(L p_{)- lim}

k Fεjk(Ax, Bρ(z)) .

We shall show the ≥ inequality, so for a simmetry argument we will have the thesis. According to proposition (5.4.1) we can take a sequence (uk) ⊂

W1,p_(B

ρ(y); Rm) converging to zero in Lp(Bρ(y); Rm) and such that

lim k Fεjk(Ax + uk(x), Bρ(y)) = Γ(L p_{)- lim} k Fεjk(Ax, Bρ(y)) . Now define τk ∈ Rn by (τk)i = εjk z_i− yi εjk .

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PERIODIC AND ALMOST PERIODIC HOMOGENIZATION

Note that τk → z − y and τk is a period for x 7→ f (x/εjk, A), so

Fε_jk(Ax + uk(x − τk), τk+ Bρ(y)) = Fε_jk(Ax + uk(x), Bρ(y)) ,

and, taken 0 < r < 1, we have definitively in k that Brρ(z) ⊂ τk+ Bρ(y). Since

uk(x − τk)|Brρ(z) belongs to W

1,p_(B

rρ(r); Rm) and tends to 0 in Lp(Brρ(z); Rm)

we have:

Γ(Lp)- lim Fε_jk(Ax, Brρ(z)) ≤ lim inf Fε_jk(Ax + uk(x − τk), Brρ(z)) ≤

≤ lim inf Fε_jk(Ax + uk(x − τk), τk+ Bρ(y)) =

= lim inf Fε_jk(Ax + uk(x), Bρ(y)) = Γ(Lp)- lim

k Fεjk(Ax, Bρ(y))

Letting r → 1 we obtain the desired inequality (remark that we have used only the inner regularity assumption).

Now given εj and a bounded open set Ω we have a subsequence εjk and a

corresponding φ. Now take the bounded open set (0, 1)n, we have that there exists a further subsequence ai of εjk and a corresponding φ2 such that

∃Γ(Lp_{)- lim}

i Fai(u, U ) =

Z

U

φ2(Du(x))dx .

for every U ∈ A((0, 1)n) and u ∈ W1,p((0, 1)n_{; R}m). Now consider the bounded open set Ω ∪ (0, 1)n; there exists a subsequence ail and a φ3 such that

∃Γ(Lp_{)- lim}

l Fail(u, U ) =

Z

U

φ3(Du(x))dx .

for every U ∈ A(Ω∪(0, 1)n_{) and u ∈ W}1,p_{(Ω∪(0, 1)}n_{; R}m_{). So, taken u(x) = Ax}

and U = Ω, we have that |U |φ3(A) = |U |φ(A), so φ3 = φ. Analogously taken

U = (0, 1)n _{we have that φ}

2 = φ3, so φ = φ2. So now it is sufficient to show

that, if Ω = (0, 1)n_{, φ is given by formula (6.3), and in particular is indipendent}

from the sequence (εj). We end the proof in the case p > 1, which is simpler.

Due to remark (2.1.6) we have φ(A) = min

Z

(0,1)n

φ(x, A + Du(x))dx : u ∈ W₀1,p((0, 1)n_{; R}m)

and by theorem (3.1.5) and (5.4.1) this equals lim k inf Z (0,1)n f ( x εjk , A + Du(x))dx : u ∈ W₀1,p((0, 1)n_{; R}m)

PietroSiorpaesProf.LuigiAmbrosio Homogenizationofintegralfunctionals

Tesi di laurea

23 giugno 2006

Homogenization

of integral

functionals

Candidato

Relatore

Contents

Chapter 1

Introduction

Chapter 2

Preliminary results

2.1

Notions of convexity

2.2

The Yosida transform

2.3

Sobolev Spaces

2.4

Almost periodic functions

Chapter 3

Γ-convergence

3.1

From Weierstrass’s theorem to Γ-

conver-gence

3.2

Γ-liminf and Γ-limsup

3.3

Properties of Γ-convergence

Chapter 4

Integral representation

4.1

An integral representation theorem

4.2

Increasing set functions

4.3

The fundamental estimate

Chapter 5

Γ-limits of functionals

5.1

Regularity of Γ-limits

5.2

Existence of Γ-limits

5.3

Representation of Γ-limits

5.4

Boundary values and Γ-limits

5.5

Γ-limits of homogeneous functionals

Chapter 6

Periodic and Almost Periodic

Homogenization

6.1

Periodic homogenization