Universit`a di Pisa
Facolt`a di Scienze Matematiche Fisiche e Naturali Corso di Laurea in Matematica
A. A. 2005-06 Elaborato finale
Homogenization
of integral
functionals
Candidato RelatoreContents
1 Γ-convergence 3
1.1 Basic facts . . . 3 1.2 Further properties . . . 5
2 Integral representation 8
2.1 An integral representation theorem . . . 8 2.2 The fundamental estimate . . . 12
3 Γ-limits of functionals 14
3.1 Existence and regularity . . . 14 3.2 Representation and boundary values . . . 17
4 Periodic and Almost Periodic Homogenization 22
4.1 Periodic homogenization . . . 22 4.2 Almost periodic homogenization . . . 24
5 Besicovitch almost periodic homogenization 27
5.1 Homogenization closure theorem . . . 27 5.2 Besicovitch almost periodic homogenization . . . 32
Chapter 1
Γ-convergence
1.1
Basic facts
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. Definition 1.1.1 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-midly-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 coercive is given by any periodic lower semicontinuous function f : Rn→ R. Given a function f : X → R, f will denote the lower semicontinuous envelope of f , given by
f = sup{g(x) : g is lower semicontinuous and g ≤ f } = f (x) ∧ lim inf
y→x f (y).
Theorem 1.1.2 If f : X → R is midly-coercive, then there exists minXf , and 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 ≤
infXf and equality holds. Let (xj) be a sequence such that xj → x and f (xj) →
infXf (if K is a compact non-empty subset of X such that infXf = infKf , then
there exists a sequence (xj) ⊆ K like that), then we have
inf
CHAPTER 1. Γ-CONVERGENCE
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. Fol-lowing the proof of theorem (1.1.2), 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, (1.1)
so if we could substitue the ellipsis by ’≤’ and if we had lim sup
j
fj(xj) ≤ inf
X f, (1.2)
then all inequalities in (1.1) and (1.2) would become equalities. Thus we are led to the following definition.
Definition 1.1.3 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); (1.3) 2. ∃ xj → x such that lim sup j fj(xj) ≤ f (x), (1.4)
or, which is equivalent by (1.3), lim
j fj(xj) = f (x). (1.5)
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 Γ(Lp). Remark 1.1.4 We observe that the definition can be given for an arbitrary topo-logical 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 1.1.5 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. (1.6)
Moreover, given a sequence (xj) such that limjfj(xj) = limjinfXfj, any limit point
x of (xj) is a minimum point for f , and there exist a sequence like that which admits
1.2. FURTHER PROPERTIES
1.2
Further properties
Now we examine the general properties of Γ-convergence. Keeping in mind what happens when we deal with limits coming from a topology, we look for analogous properties for Γ-limits. All the following properties come simply from the definitions. Remark 1.2.1 If f = Γ- limjfj then the following properties follow easily from the
definition:
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 1.2.2 The following relations between the different kind of convergences hold:
1. if fj → f uniformly then fj Γ-converges to f
2. if fj ↓ f then fj Γ-converges to f
3. if fj ↑ f then fj Γ-converges to supjfj = limjfj.
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 1.2.3 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 (1.7)
∃xj → x such that λ = lim inf
j fj(xj); (1.8)
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 1.2.4 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 (1.9)
∃xj → x such that λ = lim sup
j
CHAPTER 1. Γ-CONVERGENCE
2. λ = min{lim supjfj(xj) : xj → x}
3. λ = inf{lim supjfj(xj) : xj → x}
4. λ = sup
U ∈N (x)
lim supj inf
y∈Ufj(y).
Definition 1.2.5 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) (1.11)
if any of the equivalent conditions of proposition 1.2.3 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) (1.12)
if any of the equivalent conditions of proposition 1.2.4 is satisfied. If we have Γ(d)- lim inf
j fj(x) = λ = Γ(d)- lim supj fj(x) (1.13)
then f is called the Γ(d)- limit of (fj) at x and we write f = Γ(d)- limjfj(x). When
no confusion can arise we shall omit the dependence on the metric d.
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 1.2.6 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 than Γ(d)-lim supjfj ≤ Γ(d)-lim supjgj and
Γ(d)-lim infjfj ≤ Γ(d)-lim infjgj
3. Γ(d)-lim infj(f +g)j ≥ Γ(d)-lim infjfj+Γ(d)-lim infjgj with possible inequality
4. if fj ≥ gj+ hj than Γ(d)-lim infjfj ≥ Γ(d)-lim infjgj+ Γ(d)-lim infjhj.
We note that properties analogous to (1.2.6.(3)) and (1.2.6.(4)) are not true for the Γ(d)-lim sup. Now let us see two important properties, that we will use continuously. As for every convergence coming from a topology, we have
Theorem 1.2.7 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:
1.2. FURTHER PROPERTIES
Theorem 1.2.8 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 ∃ lim infy∈U0fσ0
j(y). We define inductively (σkj)j to be any subsequence of (σjk−1)j such that ∃ lim infy∈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 fjk(y)
so using as definition property 4 of 1.2.3) and of 1.2.4 we have that the Γ-liminf and the Γ-limsup coincide.
Chapter 2
Integral representation
2.1
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, (2.1)
with u ∈ W1,p(Ω; Rm) (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 (2.1). From now Ω, U and V will be open subsets of Rn.
Theorem 2.1.1 Let 1 ≤ p < ∞ and Mm×n be the space of m × n real matrices. If F : W1,p(Ω; Rm) × A(Ω) → [0, ∞), then there exists a quasiconvex function f : Ω × Mm×n→ [0, ∞) satisfying the growth condition
0 ≤ f (x, A) ≤ c(a(x) + |A|p) for a.e. x ∈ Ω and ∀A ∈ Mm×n (2.2) and such that
F (u, U ) = Z
U
f (x, Du(x))dx ∀u ∈ W1,p(Ω; Rm) and ∀U ∈ A(Ω) (2.3)
if and only if the following conditions hold:
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
2.1. AN INTEGRAL REPRESENTATION THEOREM
3. (translation invariance in u) F (u, U ) = F (u + z, U ) ∀z ∈ Rn, ∀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 ∈ Mm×n, y, z ∈ Ω, ρ > 0 such that B(y, ρ) ∪ B(z, ρ) ⊆ Ω . In this case there exists h : Mm×n→ [0, ∞) quasiconvex and such that
0 ≤ h(x, A) ≤ c(1 + |A|p) ∀A ∈ Mm×n
F (u, U ) =R
Uh(Du(x))dx ∀u ∈ W
1,p(Ω; Rm) and ∀U ∈ A(Ω) (2.4)
Proof One implication is obvious, the other one goes as follows: Step 1: representation on piecewise affine functions.
Fix A ∈ Mm×n, then F (Ax, ·) can be extended to a Borel measure which, by
con-dition (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)| . (2.5)
From the same theorem, if g is a measurable function that satisfies (2.2) and (2.3), then g(x, A) = f (x, A) ∀A and for a.e. x, so that function is unique up to equality x a.e. . Then by condition (2) we have (2.2). 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 \SNj=1Uj| = 0 (zero Lebegue measure),
Aj ∈ Mm×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,
CHAPTER 2. INTEGRAL REPRESENTATION
so representation (2.3) holds for such functions. Step 2: Rank one convexity of f
To show that f given by (2.5) is rank-one convex it is sufficient to show that, fixed t ∈ (0, 1) and A, B ∈ Mm×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 ∈ Wloc1,∞(Rn; Rm) and if we set uj(x) = 1jv(jx) we have Duj = A on 1jEA
and Duj = B on 1jEB and by theorem (????) that for every bounded open set V
uj ∗ * (tB + (1 − t)A)x weakly∗ in W1,∞(V ; Rm) χ1 jEA ∗ * 1 − t weakly∗ in L∞(V ) χ1 jEB ∗ * t weakly∗ in L∞(V ). So we have
F ((tB + (1 − t)A)x, B(y, ρ)) ≤ lim infjF (uj, B(y, ρ)) =
= lim infj
F (Ax,1jEA∩ B(y, ρ) + F (Bx,1jEA∩ 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)).
Then by remark (????) we have that f (y, ·) is locally Lipschitz and so f is a quasiconvex function.
Step 3: Extension of the representation by density By remark (????) the functional u 7→ R
Uf (x, Du)dx is continuous with respect to
the strong convergence in W1,p(Ω; Rm). Take U b Ω (U open and such that U is a compact subset of Ω), so that, given any u, there exists a sequence uj converging
strongly to u in W1,p(Ω; Rm) and such that u
j|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
2.1. AN INTEGRAL REPRESENTATION THEOREM
for all U b Ω and v ∈ W1,p(Ω; Rm), 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 all U ⊆ Ω (for the criterion of coincidence of measures).
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).
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 2.1.2 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 ) (2.6)
for all U and V, it is superadditive if
α(U ∪ V ) ≥ α(U ) + α(V ) (2.7)
whenever U ∩ V = ∅, and it is inner regular if
α(U ) = sup{α(V ) : V b U } (2.8)
for all U and V.
Theorem 2.1.3 (Measure property criterion) Let α : A(Ω) → [0, +∞] be an in-creasing 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 } (2.9) is a Borel measure on Ω.
CHAPTER 2. INTEGRAL REPRESENTATION
2.2
The fundamental estimate
Willing to use the theorem (2.1.1) 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 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 < ∞.
Definition 2.2.1 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 ∈ Cc∞(U ), 0 ≤ φ ≤ 1 and φ|U0 = 1.
Definition 2.2.2 F : Lp(Ω; Rn) × A(Ω) → [0, +∞] is said to satisfy the Lp -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; Rn) 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 (2.10)
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 2.2.3 If f : Ω × Rm× Rm×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) (2.11) for all x ∈ Ω, s ∈ Rm, A ∈ Rm×n.
We say that F : W1,p(Ω; Rm) × A(Ω) → [0, ∞) belongs to the class F (α, β, p) iff there exists a Borel function f : Ω × Rm× Rm×n → [0, +∞) satisfying (2.11) and
such that
F (u, U ) = Z
U
f (x, u, Du(x))dx (2.12)
for all u ∈ W1,p(Ω; Rm), U ∈ A(Ω).
In all the following when we have a functional defined for u ∈ W1,p(Ω; Rm), we shall consider that functional as defined on all Lp(Ω; Rm), meaning that it is extended to +∞ on Lp(Ω; Rm) \ W1,p(Ω; Rm) (we take this extension to mantain the property of being lower semicontinuous ).
Theorem 2.2.4 If Ω is bounded 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)).
2.2. THE FUNDAMENTAL ESTIMATE
Proof :
We have the same conclusion with the following less natural hypotesis:
Definition 2.2.5 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)). (2.13) Then F : W1,p(Ω; Rm) × A(Ω) → [0, ∞) belongs to the class F (g) iff there exists
a Borel function f : Ω × Rm× Rm×n → [0, +∞) satisfying g(x, A) ≤ f (x, s, A) ≤
c(1 + g(x, A)) and such that
F (u, U ) = Z
U
f (x, u, Du(x))dx (2.14)
for all u ∈ W1,p(Ω; Rm), U ∈ A(Ω).
Theorem 2.2.6 If Ω is bounded the family F (g) satisfies the Lp-fundamental esti-mate uniformly.
Chapter 3
Γ-limits of functionals
3.1
Existence and regularity
We are going to show a stronger Γ-compactness result, which takes into account simoutaneously the dependence of the functional on the functions and on the sets. Then from the fundamental estimate we will derive some inequalities for Γ-limits, which permit to proof that Γ-limits are subaddittive and inner regular increasing set functions. Thus in the case of integral functionals F ∈ F (α, β, p) and F ∈ F (g) we will be able to use theorem (2.1.1) to give an explicit integral representation of the Γ-limit
Theorem 3.1.1 Take 1 ≤ p ≤ +∞ and let Fj : Lp(Ω; Rm) × A(Ω) → [0, +∞] be a
sequence of functionals. Suppose that at every u ∈ A ⊆ Lp the Γ(Lp)- lim infj and
the Γ(Lp)- lim supj of the function
u 7→ Fj(u, U )
,that we will call α0(U ) and α00(U ), define inner regular increasing set functions U 7→ α0(U ) and U 7→ α00(U ). Then there exists a subsequence (jk)k such that for
all U ∈ A(Ω) the corresponding sequence of functionals u 7→ Fjk(u, U ) Γ(L
p
)-converges at all u ∈ A.
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 ). We will always apply theorem (3.1.1) taking
A = W1,p(Ω; Rm)
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 (1.2.8) there is a subsequence (σj0)j such
that ∃ Γ(Lp)- limj→+∞Fσ0
j(u, Rj). We define inductively (σ
k
j)j to be any
subse-quence of (σjk−1)j such that ∃ Γ(Lp)- limj→+∞Fσk
j(u, Rk). Then, if jk = σ
k k is the
diagonal sequence, we have that
∀R ∈ R ∃Γ(Lp)- lim
3.1. EXISTENCE AND REGULARITY
Moreover α0(R) = α00(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 infkFjk(u, U ) = α
0(U ) = sup{α0(V ) : V ∈ A(Ω), V b U } =
= sup{α0(R) : R ∈ R, R b U } = sup{α00(R) : R ∈ R, R b U } = = sup{α00(V ) : V ∈ A(Ω), V b U } = α00(U ) = Γ(Lp)- lim supkFjk(u, U ) 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 3.1.2 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 exists M, ε0 > 0 such that ∀ε < ε0, u, v ∈ Lp(Ω; Rn) 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||p (3.1) Obviously an analogous definition holds for (Fj)j∈N with j → +∞.
Theorem 3.1.3 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
Fεj(u, U ) then we have
F0(u, U0∪ V ) ≤ F0(u, U ) + F00(u, V ) (3.2) F00(u, U0∪ V ) ≤ F00(u, U ) + F00(u, V ) (3.3) for all u ∈ Lp(Ω, Rm) and U0, U, V ∈ A(Ω) with U0 b U .
Proof : Take uj, vj converging to u such that
F0(u, U ) = lim inf
j Fεj(u, U ) and F
00(u, U ) = lim sup j
Fεj(u, U )
Fix σ > 0 and U0, U, V ∈ A(Ω) with U0 b U , then apply the 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 ||u − v||p (3.4)
CHAPTER 3. Γ-LIMITS OF FUNCTIONALS
Since ||uj− vj||Lp(Ω;Rn) → 0 and ||φ||pL∞ ≤ 1 we have wj → u in Lp(Ω; Rm) and F0(u, U ∪ V ) = lim inf
j Fεj(wj, U
0∪ V ) ≤
≤ (1 + σ) lim inf
j (Fεj(uj, U ) + lim supj Fεj(vj, V )) + σ =
= (1 + σ)(F0(u, U ) + F00(u, V )) + σ
and by the arbitrariness of σ we have (3.2). The proof of (3.3) is completely analo-gous.
Theorem 3.1.4 Taken q ∈ [1, +∞), under the same hypothesis of Theorem (3.1.3) if we further suppose that Ω is bounded and that for all u ∈ W1, q(Ω; Rm)∩Lp(Ω; Rm) the limits F0(u, ·) and F00(u, ·) are increasing set functions and
F00(u, U ) ≤ c Z
U
(1 + |Du|p)dx ∀U ∈ A(Ω) (3.5) then F0(u, ·) and F00(u, ·) are inner regular increasing set functions and F00(u, ·) is subadditive.
Proof : Now take K compact contained in W ∈ A(Ω) and choose U0, U ∈ A(Ω) such that K ⊂ U0 b U b W and define V = W \ K. By (3.2) 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; Rm) ∩ Lp(Ω; Rm) by (3.5) 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 Kn such that Kn ⊆ Kn+1◦ and U =
S
nKn. Then by
(3.3) with U0 = Kn we have
F00(u, Kn◦∪ V ) ≤ F00(u, U ) + F00(u, V ) and taking the supn by the inner regularity we have
3.2. REPRESENTATION AND BOUNDARY VALUES
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 (1.2.6.(2)), and that they are super-additive by (1.2.6.(4)), combining Theorem (3.1.1), Theorem (2.1.3), and Theorem (3.1.3) we finally obtain
Theorem 3.1.5 Let Ω be bounded and (Fε)ε>0 be a family in F (α, β, p) or in F (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 ∈ W
1,p(Ω; Rm)
∃ Γ(Lp)- lim
k Fεjk(u, U ) =: F (u, U )
and F (u, ·) is the restriction of a Borel measure to A(Ω).
3.2
Representation and boundary values
Now we are able to apply the representation theorem to functionals with integrands which ‘do not depend on the variable u” to obtain
Theorem 3.2.1 Let Ω be bounded and take α, β > 0, p ∈ [1, +∞). Let (fε)ε>0 be a
family of Borel functions with f : Ω × Mm×n→ [0, +∞) satisfying the estimate α|A|p ≤ fε(x, A) ≤ β(1 + |A|p) (3.6) for all x ∈ Ω, A ∈ Mm×n, ε > 0 and let
Fε(u, U ) :=
Z
U
fε(x, Du(x))dx (3.7)
if U ∈ A(Ω) and u ∈ W1,p(Ω; Rm). Then for every sequence (εj) of positive real
numbers converging to zero there exists a subsequence (εjk) and a quasiconvex func-tion φ : Ω × Mm×n→ [0, +∞) satisfying (3.6) such that
∃ Γ(Lp)- lim
k Fεjk(u, U ) =
Z
U
φ(x, Du)dx (3.8)
for all U ∈ A(Ω) and u ∈ W1,p(Ω; Rm).
Obviously we can obtain the same result under hypothesis (2.2.5) instead of (2.11). Proof : The theorem (3.1.5) gives the existence of the Γ-limit, that we will call F (u, U ). Suppose for the moment that Ω is bounded. To apply theorem (2.1.1) we need only to prove property (4), because property (1) has been proven in theorem (3.1.5) and the other properties are obvious. Take uj * u in W1,p(Ω; Rm) and
V b Ω with Lipschitz boundary. Then, for theorem (????) applied to a minimizing subsequence, there exists a further subsequence ujk such that ujk → u in L
p(V ; Rm)
and so, for remark (1.2.1.(1)), we have F (u, V ) ≤ lim inf
CHAPTER 3. Γ-LIMITS OF FUNCTIONALS
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. So for every bounded Ω0we have found a φΩ0 : Ω0×Mm×n → [0, +∞) satisfying (3.6) and (3.8); if Ω0 and Ω00 are open bounded sets then for theorem (2.1.1) the functions φΩ0 and φΩ00 are equal x a.e. on the intersection Ω0∩ Ω00, so it is defined a function φ sactisfying the thesis on the union of all such Ω0 (so on Ω).
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 (4.2) we will consider a more general case.
If we suppose that the estimate (3.5) is satisfied uniformly, we obtain that the minimizing sequences for the Γ-limits can be taken with the same boundary value as their limits:
Theorem 3.2.2 Let Ω be bounded and take Fε: W1,p(Ω; Rm) × A(Ω) → [0, +∞]
and suppose it satisfies the Lp-fundamental estimate as ε → 0. Suppose that εj → 0+
and
Fεj(u, U ) ≤ c Z
U
(1 + |Du|p)dx (3.9)
for all U ∈ A(Ω) and u ∈ W1,p(Ω; Rm). Then if φ ∈ W1,p(Ω; Rm) and
Gφεj(u) = Fεj(u, Ω) if u ∈ φ + W 1,p 0 (Ω; Rm) +∞ otherwise we have Γ(Lp)- lim inf j Fεj(u, Ω) = Γ(L p)- lim inf j G φ εj(u) (3.10) Γ(Lp)- lim sup j Fεj(u, Ω) = Γ(L p)- lim sup j Gφεj(u) (3.11) for all u ∈ φ + W01,p(Ω; Rm)
Proof : We prove only (3.10) because (3.11) is completely analogous. One inequality is trivial, because if we take v ∈ φ + W01,p(Ω; Rm), then Γ(Lp)- lim infjFεj(v, Ω) ≤ Γ(Lp)- lim infjGφεj(v); so let us show the opposite inequality. Take σ > 0 and uj ∈ W1,p(Ω; Rm) converging to v in Lp(Ω; Rm) such that Γ(Lp)- lim infjFεj(v, Ω) = lim infjFεj(uj, Ω). Take K compact and U, U
0 ∈ A(Ω) with K ⊂ U0 b U b Ω and
define V := Ω \ K. Applying the Lp fundamental estimate as ε → 0 with u = uj we
3.2. REPRESENTATION AND BOUNDARY VALUES
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||pLp+ σ ≤ ≤ (1 + σ)(Fεj(uj, Ω) + c Z Ω\K (1 + |Dv|p)dx) + +M ||u − v||pLp+ 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; Rm) it follows that wj ∈ φ + W01,p(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:
Theorem 3.2.3 Under the hypotheses of theorem (3.2.2) and if Ω = (0, c)n and Hεj(u) = Fεj(u|Ω, Ω) if u ∈ W 1,p # ((0, c)n; Rm) +∞ otherwise then we have Γ(Lp)- lim inf j Fεj(u, Ω) = Γ(L p)- lim inf j Hεj(u) (3.12) Γ(Lp)- lim sup j Fεj(u, Ω) = Γ(L p)- lim sup j Hεj(u) (3.13) for all u ∈ W#1,p(Ω; Rm)
When the integrands do not depend on the space variable x the Γ-convergence reduces to pointwise convergence:
Theorem 3.2.4 Take α, β > 0, p ∈ (1, +∞) and Ω ∈ A(Rn). Let (εj) be a sequence
of positive real numbers converging to zero and (fε)ε>0 be a family of continuous
functions with fε: Mm×n→ [0, +∞) satisfying the estimate
α|A|p≤ fε(A) ≤ β(1 + |A|p) (3.14)
for all A ∈ Mm×n, ε > 0 and let Fε(u, U ) :=
Z
U
fε(Du(x))dx. (3.15)
for any open set U ⊆ Ω and for all u ∈ W1,p(U ; Rm).
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; Rm) we have ∃ Γ(Lp)- lim j Fεj(u, B) = Z B f (Du(x))dx . (3.16)
CHAPTER 3. Γ-LIMITS OF FUNCTIONALS
In this case f is a quasiconvex function which satisfies the standard growth estimate of order p and for any open set U ⊆ Ω and any u ∈ W1,p(U ; Rm) we have
∃ Γ(Lp)- lim j Fεj(u, U ) = Z U f (Du(x))dx . (3.17) .
Proof : Applying theorem (3.2.1) and condition 5 of theorem (2.1.1) 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 (3.18)
at all u ∈ W1,p(U ; Rm) 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 RV f (Du(x))dx and thus by (1.2.7) we have one implication and that equality (3.17) holds.
Chose a A ∈ Mm×n and an open ball B b U , then by remark (???) we have |B|φ(A) = inf{ Z B φ(Du(x))dx : u ∈ Ax + W01,p(U ; Rm)} (3.19) |B|Qfjk(A) = inf{ Z B fjk(Du(x))dx : u ∈ Ax + W01,p(U ; Rm)} ; (3.20)
by theorem (1.1.5) and (3.2.2) we have ∃ lim k inf{ Z B fjk(Du(x))dx : u ∈ Ax + W01,p(U ; Rm)} = inf{ Z B φ(Du(x))dx : u ∈ Ax + W01,p(U ; Rm)}
so with equations (3.19) and (3.20) we obtain |B|φ(A) = limk|B|Qfjk(A) = |B|f (A).
The other implication is similar: for every bounded open set U ⊆ Ω, for every subse-quence there exists a further subsesubse-quence (εjk)k such that (3.18), so the hypothesis implies R
Bf (Du(x))dx =
R
Bφ(Du(x))dx and so f = φ; so f is quasiconvex . The
preceeding arguments then give |B|f (A) = inf{ Z B f (Du(x))dx : u ∈ Ax + W01,p(U ; Rm)} = (3.21) = lim k inf{ Z B fjk(Du(x))dx : u ∈ Ax + W01,p(U ; Rm)} = lim k |B|Qfjk(A) (3.22)
so we obtain ∃ limk|B|Qfjk(A) = |B|f (A) and the thesis.
3.2. REPRESENTATION AND BOUNDARY VALUES
Corollary 3.2.5 Let f : Mm×n→ [0, +∞) be a continuous functions satisfying the estimate
α|A|p ≤ f (A) ≤ β(1 + |A|p) (3.23) for all A ∈ Mm×n and with α, β > 0, p ∈ (1, +∞). Let
F (u, U ) := Z
U
f (Du(x))dx. (3.24)
for all U ∈ A(Ω) and for all u ∈ W1,p(U ; Rm). If F is the lower semicontinuous envelope with respect to the Lp-topology of the functional F , then
F (u) = Z
U
Qf (Du(x))dx for all U ∈ A(Ω) and for all u ∈ W1,p(U ; Rm).
Chapter 4
Periodic and Almost Periodic
Homogenization
4.1
Periodic homogenization
We shall see that, under periodicity hypothesis, not only some sequences of func-tionals Fεj(u, U ) Γ-converge to a functional of the form F (u) =
R
Uf (Du(x))dx, but
the whole family do that.
The definition of Γ-convergence for families (Fε) is completely analogous to the
de-finition for sequences (1.1.3), 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
(1.2.8) we have that the whole family converges to λ at u if and only if every se-quence admits a subsese-quence along which the family converges to λ at u. So applying theorem (3.2.1) we need only to show that the function φ(x, Du) is indipendent from x and from the sequence (εj).
In what follows we take
Fε(u, U ) =
Z
U
f (x
ε, Du(x))dx (4.1)
whenever U ∈ B(Ω) and we suppose Ω = Rn, p ∈ [1, +∞) and that f : Rn×Mm×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 ∈ Mm×n, (4.2) i.e. that f (x + ei, A) = f (x, A) for all x ∈ Rn, A ∈ Mm×n, i ∈ {1, . . . , n}.
Observe that in the case p > 1 we will obtain a complete description of the Γ-limit because, due to the standard growth condition, there exists the Γ-limit on Lp\ W1,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
4.1. PERIODIC HOMOGENIZATION
Theorem 4.1.1 In the previous hypothesis there exists a (unique) quasiconvex func-tion fhom: Mm×n→ [0, +∞) which satisfies the standard growth condition of order
p and such that for every bounded open set Ω ⊂ Rn and at every u ∈ W1,p(Ω; Rm) 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; Rm) ) = fhom(A) . (4.3) Proof : The proof of the existence of limit (4.3) is obtained by taking, with fixed t, a function utapprossimating the inf at t, and finding an approssimation usof 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 ∈ ZN.
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 (3.2.2) we can take a sequence (uk) ⊂ W01,p(Bρ(y); Rm)
converging to zero in Lp(Rn; Rm) (we take the extension which is zero out of Bρ(y))
and such that lim
k Fεjk(Ax + uk(x), Bρ(y)) = Γ(L p)- lim
k Fεjk(Ax, Bρ(y)) .
Now define τk∈ Rn and vk∈ W01,p(Rn; Rm) by
(τk)i= εjk
zi− yi
εjk
vk(x) = uk(x − τk) .
Note that τk→ z − y and τk is a period for x 7→ f (x/εjk, A), so Fεjk(Ax + vk(x), τk+ Bρ(y)) = Fεjk(Ax + uk(x), Bρ(y)) ,
and, taken r > 1, we have definitively in k that τk+ Bρ(y) ⊂ Brρ(z). But vk = 0
out of τk+ Bρ(y), so we have:
Γ(Lp)- limkFεjk(Ax, Bρ(z)) ≤ Γ(Lp)- limkFεjk(Ax, Brρ(z)) ≤
≤ lim inf Fε
jk(Ax + vk(x), Brρ(z)) ≤
≤ lim inf Fεjk(Ax + uk(x), Bρ(y)) + |Brρ\ Bρ|β(1 + |A|p) =
CHAPTER 4. PERIODIC AND ALMOST PERIODIC HOMOGENIZATION
Letting r → 1 we obtain the desired inequality.
To conclude we need to show that φ is indipendent from the sequence (εj). We end
the proof in the case p > 1, which is simpler. Due to remark (????) we have
φ(A) = min ( Z (0,1)n φ(x, A + Du(x))dx : u ∈ W01,p((0, 1)n; Rm) )
and by theorem (1.1.5) and (3.2.2) this equals lim k inf ( Z (0,1)n φ( x εjk , A + Du(x))dx : u ∈ W01,p((0, 1)n; Rm) )
which is, taken Tk= 1/εjk and with a change of variable ,
lim k inf ( 1 Tn k Z (0,Tk)n φ(x, A + Du(x))dx : u ∈ W01,p((0, Tk)n; Rm) ) ,
so by the existence of the limit (4.3) we have the thesis. This also gives (4.3). Remark 4.1.2 In the hypothesis of theorem (4.1.1) there exists
lim j→+∞ 1 jninf ( Z (0,j)n f (x, A + Du(x))dx : u ∈ W#1,p((0, j)n; Rm) ) (4.4)
and it equals fhom, where fhom is the function defined on theorem (4.1.1).
In the case of p > 1 this follows simply applying the same reasoning as the final part of the proof of theorem (4.1.1), using remark (???) instead of remark (???).
4.2
Almost periodic homogenization
We want to generalize the previous results to integrands of the form f (x, s, A). Even if we suppose the function f to be 1-periodic in the first to variables, the functions x 7→ f (x, Ax, A) are generally not periodic and this suggest it would be better not to study the periodic case. We shall see that we do not need the whole strenght of the periodicity assumption: working with almost periods (instead of periods) we will obtain the same inequalities and the same conclusions (provided that p > 1). So let us consider the family of functionals
Fε(u, U ) = Z U f (x ε, u(x) ε , Du(x))dx (4.5)
with U b Rn. It is reasonable to expect that its Γ-limit for ε → 0 is translational invariant in u, so that we will be able to apply the representation theorem.
4.2. ALMOST PERIODIC HOMOGENIZATION
function which satisfies the standard growth condition of order p and that the sets TηA and Tη, that we are going to define, are relatively dense in Rn (see def(????)).
TηAis the set of all the τ ∈ Rn such that for all x ∈ Rn, s ∈ Rm, Y ∈ Mm×n we have |f (x + τ, s + Aτ, Y ) − f (x, s, Y )| < η(1 + |Y |p) , (4.6) and Tη is the set of all the τ ∈ Rn such that for all x ∈ Rn, s ∈ Rm, Y ∈ Mm×n we
have
|f (x, s + τ, Y ) − f (x, s, Y )| < η(1 + |Y |p). (4.7) Everything works has hoped and we obtain the following
Theorem 4.2.1 In the previous hypothesis there exists a quasiconvex function fhom:
Mm×n → [0, +∞) which satisfies the standard growth condition of order p and such that for every bounded open set Ω ⊂ Rn and at every u ∈ W1,p(Ω; Rm) we have
∃Γ(Lp)- lim
ε→0Fε(u, Ω) =
Z
Ω
fhom(Du(x))dx .
Moreover there exists lim t→+∞ 1 tninf ( Z (0,t)n f (x, Ax + u(x), A + Du(x))dx : u ∈ W01,p((0, t)n; Rm) ) (4.8)
and it equals fhom(A).
Proof : Now take a sequence εj of positive real numbers converging to
zero, than theorem (3.1.5) gives the existence along a subsequence (εjk) of the Γ-limit, that we will call F (u, U ). Now fix a bounded open set U , a a ∈ Rn and a u ∈ W1,p(U ; Rm). If we show that F (u + a, U ) = F (u, U ), we can apply the representation theorem and get for every εjk a φ(x, Du); then we shall conclude, as in theorem (4.1.1), proving φ to be independent from x and from the sequence εjk. So let us show F (u + a, U ) ≤ F (u, U ), which applied to u − a instead of u gives the equality. Fix η > 0 and take (uk) ⊂ W1,p(U ; Rm) converging in Lp to u and such
that F (u, U ) = lim k Z U f ( x εjk ,uk(x) εjk , Duk(x))dx.
There exists ak→ a such that τk:= ak/jk ∈ Tη, so we have F (u + a, U ) ≤ lim inf k Z U f ( x εjk ,uk(x) + ak εjk , Duk(x))dx ≤ ≤ lim k Z U f ( x εjk ,uk(x) εjk , Duk(x))dx + η lim inf k (|U | + ||Du|| p Lp(U ;Mm×n)) ≤ ≤ F (u, U ) + η(|U | + F (u, U )/α)
CHAPTER 4. PERIODIC AND ALMOST PERIODIC HOMOGENIZATION
and letting η → 0 we have the desired inequality. The proof that φ is indipendent from x is analogous to the proof in the periodic case: working with almost periods instead of periods we obtain a similar inequality. As in the periodic case we have
φ(A) = lim k inf ( 1 Tkn Z (0,Tk)n φ(x, A + Du(x))dx : u ∈ W01,p((0, Tk)n; Rm) ) , (4.9)
so the existence of the limit (4.8) would give the indipendence of φ from εj. The
proof of existence is obtained in a similar way to theorem (4.1.1), working with al-most periods instead of periods.
As in the periodic case it exists the Γ-limit at every u ∈ Lp\W1,pat it equals +∞,
since this was a consequence of the coercivity assumption, not of the periodicity’s one. Moreover there exists
lim j→+∞ 1 jninf ( Z (0,j)n f (x, Ax + u(x), A + Du(x))dx : u ∈ W#1,p((0, j)n; Rm) ) (4.10) and it equals fhom: this follows simply applying the same reasoning as the final part
Chapter 5
Besicovitch almost periodic
homogenization
5.1
Homogenization closure theorem
We shall show that homogenization is stable under some kind of perturbation, and this will permit us to achieve a strong homogenization theorem, which holds under very mild assumption of regularity.
To start we need a technical tool, namely the Yosida transform:
Definition 5.1.1 Let (X, d) be a metric space, ψ : [0, +∞) → [0, +∞) a strictly increasing continuous function with ψ(0) = 0, λ ≥ 0 and f : X → R. We define the Yosida transform of f , Tλψf : X → R by
Tλψf (x) = inf{f (y) + λψ(d(x, y)) : y ∈ X}. (5.1) If ψ(t) = tp we write simply Tλpf .
We remark that Tλψf (x) is increasing in λ and in f . The Yosida transform will be useful to us because of the following theorem:
Theorem 5.1.2 Let (X, d) be a metric space, ψ : [0, +∞) → [0, +∞) a strictly increasing continuous function with ψ(0) = 0 and fj : X → R a sequence. We have
for all x
Γ(d)- lim inf
j fj(x) = limλ→+∞lim infj T ψ λfj(x) (5.2) Γ(d)- lim sup j fj(x) = lim λ→+∞lim supj T ψ λfj(x) . (5.3)
The proof is straightforward.
We need also the following partial regularity result due to Meyers, that we will not prove:
CHAPTER 5. BESICOVITCH ALMOST PERIODIC HOMOGENIZATION
Theorem 5.1.3 Take p > 1 and let f : Rn× Mm×n→ R be a quasiconvex function
satisfying the standard growth estimate of order p. Fix an open ball B ⊂ Rn and a φ ∈ C∞(B; Rm). Then there exist η > 0 such that, for all λ ≥ 0, there exists C > 0 such that the minimum points wλ on W01,p(B; Rm) of the functional
Gλ(w) = Z B f (x, Dw(x) + Dφ(x)) + λ|w(x)|pdx (5.4) belong to W1,p+η(B; Rm) and ||wλ||W1,p+η(B;Rm)≤ C (5.5) Let us define what we will mean by ‘perturbation”
Definition 5.1.4 Let (fj) and (φj) be two sequences of real Borel fuctions defined
on Rn× Mm×n. We say that they are equivalent sequences if we have, for all y ≥ 0,
lim j lim supT →+∞ 1 (2T )n Z (−T,T )n sup |A|≤y |fj(x, A) − φj(x, A)|dx = 0 . (5.6)
We remark that we have just defined an equivalence relation, and that if (fj) and
(φj) are equivalent sequences, then for every bounded open set Ω we have
lim j lim sup→0 Z Ω sup |A|≤y |fj(x/, A) − φj(x/, A)|dx = 0 . (5.7)
So let us see that some limits involving Yosida transforms are not changed by per-turbations.
Theorem 5.1.5 Let (fj) and (φj) be equivalent sequences of quasiconvex funtions
uniformly satisfying the standard growth condition of order p, with p > 1. Take an open ball B ⊂ Rn and φ ∈ C∞(B; Rm) and define the functionals
Fj(u, B) = Z B fj( x , Du(x))dx , Φ j (u, B) = Z B φj( x , Du(x))dx (5.8) for all u ∈ φ + W1,p(B; Rm), and extended to +∞ elsewhere in Lp(B; Rm).Then
lim inf j lim sup→0+ T p λF j (φ, B) = lim inf j lim sup→0+ T p λΦ j (φ, B) (5.9) lim sup j lim sup →0+ TλpFj(φ, B) = lim sup j lim sup →0+ TλpΦj(φ, B) (5.10) Proof : Let us prove (5.9), (5.9) having an analogous proof; for a simmetry argument it suffices to prove an inequality. By definition we have that TλpΦj(φ, B)
equals min Z B φj(x/, Dw(x) + Dφ(x)) + λ|w(x)|pdx : w ∈ W01,p(B; Rm) ,
5.1. HOMOGENIZATION CLOSURE THEOREM
hence by theorem (5.1.3) there exist η and C(λ) such that ||wλ+ φ||W1,p+η(B;Rm)≤ C(λ) .
Now define gj(x, y) = sup{|fj(x, A) − φj(x, A)| : |A| ≤ y} and observe that y 7→
gj(x, y) are increasing and that 0 ≤ gj(x, y) ≤ 2β(1 + yp). Thus we can estimate as
follows, taken BK := {x ∈ B : Dwλ(x) + Dφ(x)) > K}: TλpFj(φ, B) ≤ Z B fj(x/, Dwλ+ Dφ) + λ|wλ|pdx ≤ ≤ TλpΦj(φ, B) + Z B |fj(x/, Dwλ+ Dφ) − φj(x/, Dwλ+ Dφ)|dx ≤ ≤ TλpΦj(φ, B) + Z B\BK gj(x/, Dwλ+ Dφ)dx + 2β Z BK (1 + |Dwλ+ Dφ|p)dx ≤ ≤ TλpΦj(φ, B) + Z B gj(x/, K)dx + 2β |BK| + |BK|η+pη ( Z B |Dwλ+ Dφ|η+pdx)η+pp . Now let us estimate |BK|:
|BK|Kp = Z BK Kpdx ≤ Z BK |Dwλ+ Dφ|pdx ≤ Z B |Dwλ+ Dφ|pdx ≤ ≤ 1 α Z B φj(x/, Dwλ+ Dφ)dx ≤ 1 αT p λΦ j ≤ ≤ 1 α Z B φj(x/, Dφ)dx ≤ β α Z B (1 + |Dφ|p)dx)(φ, B) ,
so there exists a constant C0 such that |BK| ≤ C0Kp. Taking first the lim sup→0,
after the lim infj and finally the limit as K → +∞ we obtain for every λ ≥ 0
lim inf j lim sup→0+ T p λF j (φ, B) ≤ lim inf j lim sup→0+ T p λΦ j (φ, B).
We are now able to prove the Homogenization closure theorem, which states that ho-mogenization is stable under some very general perturbations, by combing theorem (5.1.2) and theorem (5.1.5):
Theorem 5.1.6 Let p > 1, let f and φj be quasiconvex functions defined on Rn×
Mm×n and satisfying uniformly the standard growth condition of order p. Suppose moreover that (φj) is equivalent to the constant sequence (f ). If for all j ∈ N there
exists a quasiconvex function ψj such that for all open bounded sets Ω ⊂ Rn and at
all u ∈ W1,p(Ω; Rm) we have ∃Γ(Lp)- lim →0+ Z Ω φj(x/, Du(x))dx = Z Ω ψj(Du(x))dx (5.11)
(the functionals are extended to +∞ at Lp\ W1,p) then at every A ∈ Mm×n there
exists limj ψj(A) =: fhom(A), fhom is a quasiconvex function and for all open
bounded sets Ω ⊂ Rn and at all u ∈ W1,p(Ω; Rm) we have ∃Γ(Lp)- lim →0+ Z Ω f (x/, Du(x))dx = Z Ω fhom(Du(x))dx . (5.12)
CHAPTER 5. BESICOVITCH ALMOST PERIODIC HOMOGENIZATION Moreover ∃ lim t→+∞ 1 tninf ( Z (0,t)n f (x, A + Du(x))dx : u ∈ W01,p((0, t)n; Rm) ) = fhom(A) (5.13) As in the periodic case it exists the Γ-limit at every u ∈ Lp\ W1,p at it equals +∞.
Moreover there exists lim j→+∞ 1 jninf ( Z (0,j)n f (x, A + Du(x))dx : u ∈ W#1,p((0, j)n; Rm) ) (5.14) and it equals fhom.
Proof : Fix a bounded open set Ω ⊂ Rn, by theorem (3.2.1) for every k → 0
there exists a subsequence, which we still denote by k, such that
∃Γ(Lp)- lim k
Z
U
f (x/k, Du(x))dx =: F (u, U ) ,
at every u ∈ W1,p(Ω; Rm) and for every U ∈ A(Ω) and there exists a quasiconvex function ψ which satisfies the standard growth condition of order p and such that R
Uψ(x, Du(x))dx = F (u, U ) for every u ∈ W1,p(Ω; Rm) and for every U ∈ A(Ω).
Now chose a φ ∈ W1,p(Ω; Rm) and take Fε(u, Ω) := R Ωf (x/, Du(x))dx if u ∈ φ + W 1,p 0 (Ω; Rm) +∞ otherwise , (5.15)
then by theorem (3.2.2) and theorem (5.1.2), extending to +∞ the functional F (·, Ω) to Lp\ W1,p, we have F (u, Ω) = lim λ→+∞lim infk T p λFεk(u, Ω) = lim λ→+∞lim supk T p λF φ εk(u, Ω) (5.16) at every u ∈ φ + W01,p(Ω; Rm). Analogously define
Φj(u, Ω) := R Ωφj(x/ε, Du(x)) if u ∈ φ + W 1,p 0 (Ω; Rm) +∞ otherwise , Ψj(u, Ω) := R Ωψj(Du(x)) if u ∈ φ + W 1,p 0 (Ω; Rm) +∞ otherwise ,
then by theorem (3.2.2) we have Ψj(u, Ω) = Γ- lim→0Φj(u, Ω) at every u ∈ Lp(Ω; Rm).
Now take an open ball B b Ω and let us prove that TλpΨj(φ, B) = lim →0+ T p λΦ j ε(φ, B) , (5.17)
or equivalently that the equality holds for every sequence k→ 0+. So fix a sequence
5.1. HOMOGENIZATION CLOSURE THEOREM
(Φjk(u, B))kis equi-mildly coercive on L
p , so theorem (1.1.5) and remark (1.2.1.(3))
with fk(u) = Φjk(u, B) and with g(u) = λ R B|u(x) − φ(x)| pdx gives that inf{Ψj(u, B) + λ Z B |u(x) − φ(x)|p : u ∈ W1,p(B; Rm)} = = lim k inf{Φ j k(u, B) + λ Z B |u(x) − φ(x)|p : u ∈ W1,p(B; Rm)}
that is the desired equality. Now take the subsequence k chosen at the beginning
of this proof, then equation (5.17) and theorem (5.1.2) and (5.1.6) with fj = f give
Γ(Lp)- lim inf j Ψ j(φ, B) = lim λ→+∞lim infj T p λΨ j(φ, B) = lim
λ→+∞lim infj →0+lim T p λΦ
j
(φ, B) = lim
λ→+∞lim infj lim sup→0+ T p
λF(φ, B) =
= lim
λ→+∞lim sup→0+ T p
λF(φ, B) ≥ limλ→+∞lim sup k TλpFk(φ, B) ≥ ≥ lim λ→+∞lim infk T p λFk(φ, B) ≥ lim λ→+∞lim inf→0+ T p λF(φ, B) = = lim
λ→+∞lim supj lim inf→0+ T p
λF(φ, B) = limλ→+∞lim sup j lim inf →0+ T p λΦ j (φ, B) = = lim λ→+∞lim supj T p λΨ j(φ, B) = Γ(Lp)- lim sup j Ψj(φ, B) so there exists Γ(Lp)- lim
jΨj(φ, B) for any open ball B b Ω. Then by theorem
(3.2.2), defined Ψj(u, V ) :=
R
V ψj(Du(x)) if u ∈ W1,p(V ; Rm), V ∈ A(Rn)
+∞ otherwise ,
there exists Γ(Lp)- limjΨ j
(φ, B) and it equals Γ(Lp)- limjΨj(φ, B). So, by equation
(5.16), the Γ(Lp)- limjΨ j
(φ, B) equals F (φ, B), so by theorem (3.2.4) there exists lim ψj(x, A) = ψ(A) and for every bounded open set V there exists Γ(Lp)- limjΨ
j
(φ, V ) and it equals F (φ, V ). So F does not depende on the sequence (k) and ψ does not
depend on x and we have the thesis. The final part of the proof of theorem (4.1.1) can be applied to give equality (5.13).
Corollary 5.1.7 Let p > 1, let f and φ be quasiconvex functions defined on Rn×
Mm×n and satisfying the standard growth condition of order p. Suppose moreover that for all y ∈ R
lim sup T →+∞ 1 (2T )n Z (−T,T )n sup |A|≤y |f (x, A) − φ(x, A)|dx = 0
and that there exists a quasiconvex function φhomsuch that for all open bounded sets
Ω ⊂ Rn and at all u ∈ W1,p(Ω; Rm) we have ∃Γ(Lp)- lim →0+ Z Ω φ(x/, Du(x))dx = Z Ω φhom(Du(x))dx . (5.18)
CHAPTER 5. BESICOVITCH ALMOST PERIODIC HOMOGENIZATION
Then for all open bounded sets Ω ⊂ Rn and at all u ∈ W1,p(Ω; Rm) ∃Γ(Lp)- lim →0+ Z Ω f (x/, Du(x))dx = Z Ω φhom(Du(x))dx (5.19) and ∃ lim t→+∞ 1 tninf ( Z (0,t)n f (x, A + Du(x))dx : u ∈ W01,p((0, t)n; Rm) ) = φhom(A) (5.20) Condition (5.1.7) is satisfied if φ(x, A) − f (x, A) is a continuous function such that for every A the function x 7→ φ(x, A) − f (x, A) has compact support; in this sense Homogenization is stable under compact support perturbations.
5.2
Besicovitch almost periodic homogenization
Now we give a much weaker definition of almost periodic function, namely the Besi-covicth’s one; thanks to the homogenization theorem for almost periodic functions and to the homogenization closure theorem, we prove by approssimation a homoge-nization theorem for Besicovicth almost periodic functions.
Definition 5.2.1 A measurable function v : Rn→ R is Besicovicth almost periodic if there exists a sequence of trigonometric polynomials Pk(x) =
Prk j=1akj exp (i(λkj, x)) such that lim sup T →+∞ 1 (2T )n Z (−T,T )n |Pk(x) − v(x)| dx = 0 . (5.21)
Theorem 5.2.2 Suppose that p > 1 and f : Rn×Mm×n → [0, +∞) is a quasiconvex
function satisfying the standard growth condition of order p and such that for every A ∈ Mm×n the function x 7→ f (x, A) is Besicovitch almost periodic. Then there exists a quasiconvex function fhom : Mm×n → [0, +∞) which satisfies the standard
growth condition of order p and such that for every bounded open set Ω ⊂ Rn and at every u ∈ W1,p(Ω; Rm) we have ∃Γ(Lp)- lim ε→0 Z Ω fε(x/ε, Du(x))dx = Z Ω fhom(Du(x))dx .
Moreover there exists lim t→+∞ 1 tninf ( Z (0,t)n f (x, A + Du(x))dx : u ∈ W01,p((0, t)n; Rm) ) (5.22)
5.2. BESICOVITCH ALMOST PERIODIC HOMOGENIZATION
As in the periodic case it exists the Γ-limit at every u ∈ Lp\ W1,p at it equals +∞.
Moreover there exists lim j→+∞ 1 jninf ( Z (0,j)n f (x, A + Du(x))dx : u ∈ W#1,p((0, j)n; Rm) ) (5.23) and it equals fhom.
Proof : We shall construct a sequence of almost periodic functions φkwhich satisfy
the standard growth condition of order p uniformly and prove it is equivalent to the constant sequence f : then the closure theorem, together with theorem (4.2.1), gives the thesis.
Let Aj be a dense sequence in Mm×n. For every k ∈ N, j ∈ {1, . . . , k} chose a
trigonometric polynomial Pkj(x) such that lim sup T →+∞ 1 (2T )n Z (−T,T )n |Pkj(x) − f (x, Aj)| dx ≤ 1 k2 (5.24) and define fk(x, A) = ( fk(x, A) if A ∈ {A1, . . . , Ak} β(1 + |A|p) otherwise (5.25) ψk(x, A) = ( (Pkj(x) ∨ α|A|p) ∧ β(1 + |A|p) if A ∈ {A1, . . . , Ak} β(1 + |A|p) otherwise . (5.26)
Moreover define φk := Qψk and remark that ψk satisfies the standard growth
con-dition of order p with the same α and β of f . The functions A 7→ α|A|p and
A 7→ β(1 + |A|p) are convex and hence quasiconvex, so by theorem (???) we have that also φksatisfies the standard growth condition of order p with the same α and
β of f .
Let us prove that φkis almost periodic; since it does not depend on the variable “u”
it will be sufficient to show that, given η > 0, the set of the τ such that |φk(x + τ, A) − φk(x, A)| ≤ η(1 +
β
α(1 + |A|
p)) (5.27)
is relatively dense. Since the trigonometric polynomials Pkj(x) and the constant functions are almost periodic functions, and since the set of almost periodic functions is clearly a vector space closed under lattice operations, the functions (x, s, A) 7→ ψk(x, Aj) have a relatively dense set Tη such that for every τ ∈ Tη we have
|ψk(x + τ, Aj) − ψk(x, Aj)| ≤ η(1 + |Aj|p) .
This conditions trivially extends to ψk(x, A) for general A. For any open ball B, for
every u ∈ W01,p(B; Rm) and for every τ ∈ Tη we have
φk(x + τ, A) ≤ 1 |B| Z B ψk(x + τ, A + Du(x)) dx ≤ 1 |B| Z B ψk(x, A + Du(x)) + η(1 + |Du(x) + A|p)dx
CHAPTER 5. BESICOVITCH ALMOST PERIODIC HOMOGENIZATION
now taking a minimizing sequence (ui) for the
R Bψk(x, A + Du(x))dx, observing that lim sup i→+∞ α |B| Z B |Dui+ A|pdx ≤ lim i→+∞ 1 |B| Z B ψk(x, Dui+ A)dx = φk(x, A) ≤ β(1 + |A|p) we finally obtain φk(x + τ, A) − φk(x, A) ≤ η(1 + β α(1 + |A| p)); (5.28)
by a simmetry argument then we obtain (5.27).
Now observe that, by the quasiconvexity of f and the fact that f ≤ fkand f (x, Aj) =
fk(x, Aj), we have that f ≤ Qfk≤ fk and Qfk(x, Aj) = f (x, Aj) for all j = 1, ..., k.
But f (x, A) and Qfk(x, A) are positive quasiconvex functions bounded by β(1+|A|p),
so by remark (???) there exists a constant c such that
|f (x, A) − f (x, B)| ≤ c(1 + yp−1)|A − B| whenever |A|, |B| ≤ y
|Qfk(x, A) − Qfk(x, B)| ≤ c(1 + yp−1)|A − B| whenever |A|, |B| ≤ y
But we can estimate
|f (x, A) − Qfk(x, A)| ≤ |f (x, A) − f (x, Ai)| + |f (x, Ai) − Qfk(x, Ai)| +
+|Qfk(x, Ai) − Qfk(x, A)| = |f (x, A) − f (x, Ai)| + |Qfk(x, Ai) − Qfk(x, A)|
so we obtain, for every A such that |A| ≤ y, that |f (x, A) − Qfk(x, A)| ≤ 2c(1 + |y|p−1) inf
j {|A − Aj| : |Aj| ≤ y, j = 1, ..., k} . By density of {Aj} we have lim k lim supT →+∞ 1/(2T ) n Z (−T,T )n sup |A|≤y inf j {|A − Aj| : |Aj| ≤ y, j = 1, ..., k} = 0 ,
so we obtain that the sequence (Qfk) is equivalent to the constant sequence (f ).
To conclude we just need to show that (Qfk) is equivalent to (φk). Take an open ball
B and a minimizing sequence (ui) for the
R
Bψk(x, A + Du(x))dx; by the equality
ψk(x, Aj) = α|A|p if Pkj(x) ≤ α|A|p Pkj(x) if α|A|p≤ Pkj(x) ≤ β(1 + |A|p) β(1 + |A|p) if β(1 + |A|p) ≤ Pkj(x) we obtain Qfk(A) ≤ 1 |B| Z B fk(x, A + Dui(x)) dx ≤ (5.29) sup x∈B |fk(x, ·) − ψk(x, ·)| + 1 |B| Z B ψk(x, A + Dui(x)) dx ≤ (5.30) k X j=1 |f (x, Aj) − Pkj(x)| + 1 |B| Z B ψk(x, A + Dui(x)) dx (5.31)
5.2. BESICOVITCH ALMOST PERIODIC HOMOGENIZATION
so taking the limi→+∞ and applying a simmetry argument we obtain
|Qfk(x, A) − ψk(x, A)| ≤ k
X
j=1
|f (x, Aj) − Pkj(x)|
