[PPT] - Homogenization of a quasilinear elliptic problem with nonlinear PowerPoint Presentation

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Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions

Patrizia Donato

University of Rouen

Symposium Homogenization and Multi Scale Analysis Shanghai, October 3-7, 2011

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Outline

1

Position of the problem

2

Existence and uniqueness of a solution

3

Homogenization results

4

The quasilinear homogenized matrix

5

Uniqueness of the homogenized problem

6

About the unfolding method

7

Some preliminary results

8

Proofs of the homogenization results

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Results in collaboration with Bituin Cabarrubias, Ph.D. student under joint tutorage, University of Philippines and University of Rouen, in

B. Cabarrubias and P. Donato, Existence and Uniqueness for a

Quasilinear Elliptic Problem With Nonlinear Robin Conditions, to appear in Carpathian J. Math., (2) 2011.

B. Cabarrubias and P. Donato, Homogenization of a quasilinear

elliptic problem with nonlinear Robin boundary conditions, Applicable Analysis, 2011, to appear.

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The problem

Aim: To study the homogenization, as ε → 0, of the following quasilinear elliptic problem:        − div(Aε(x, uε)∇uε) = f in Ω∗

ε,

uε = 0

n Γε

0,

Aε(x, uε)∇uε · n + εγτε(x)h(uε) = gε(x)

n Γε

1,

where Ω∗

ε is a periodically perforated domain of RN, N ≥ 2,

∂Ω∗

ε is decomposed in two disjoint parts, Γε 0 and Γε 1,

n is the unit exterior normal to Ω∗

ε,

γ is a real parameter, with γ ≥ 1.

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Position of the problem

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The perforated domain

We denote by b = (b1, b2, . . . , bN) a basis of RN, N ≥ 2, Y the corresponding reference cell, i.e. Y = {y ∈ RN| y = N

i=1 yibi, (y1, . . . , yN) ∈ (0, 1)N},

S (the reference hole) a compact subset of Y , Y ∗ = Y \S the perforated reference cell, {ε} be a positive sequence that converges to zero, G = {ξ ∈ RN|ξ = N

i=1 kibi, (k1, . . . , kN) ∈ Z

ZN} , Ω a bounded open set of RN. The perforated domain Ω∗

ε is defined by

Ω∗

ε = Ω \ Sε,

where Sε =

ξ∈G

ε(ξ + S).

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Position of the problem

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Figure 1. The perforated domain Ω∗

ε.

We assume that S has a Lipschitz continuous boundary with a finite number of connected components.

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The two components of the boundary

As in the unfolding periodic method, we set

Ωε = interior{
ξ∈Ξε

ε(ξ + ¯ Y )} and Λε = Ω\ Ωε, where Ξε = {ξ ∈ G, ε(ξ + Y ) ⊂ Ω} . By construction, Ωε is the interior of the largest union of ε(ξ + ¯ Y ) cells such that ε(ξ + Y ) ⊂ Ω, while Λε is the subset of Ω from the ε(ξ + ¯ Y ) cells intersecting ∂Ω. We consider the corresponding perforated sets

Ω∗

ε =

Ωε \ Sε and Λ∗

ε = Ω∗ ε\

Ω∗

ε

and we decompose the boundary of the perforated domain Ω∗

ε as

∂Ω∗

ε = Γε 0 ∪ Γε 1,

where Γε

1 = ∂

Ω∗

ε ∩ ∂Sε

and Γε

0 = ∂Ω∗ ε \ Γε 1,

so that Γε

1 is the boundary of the set of holes included in

Ωε.

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Position of the problem

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Figure 2. The perforated domain Ω∗

ε and its boundary ∂Ω∗ ε = Γε 0 ∪ Γε 1.

In the figure, Ωε is the dark perforated part, Γε

1 is the boundary of the holes contained in

Ωε, Λ∗

ε is the remaining perforated part,

Γε

0 is the union of ∂Ω and the boundary of the holes in Λ∗ ε.

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Assumptions on the data

Concerning the equation − div(Aε(x, uε)∇uε) = f , we assume that f ∈ L2(Ω), A : Y × R − → RN2 is a matrix field s.t.                                        (i) A(y, t) is Y - periodic for every t and a Caratheodory function, i.e. continuous for a.e. y and measurable for every t; (ii)A(·, t) ∈ M(α, β, Y ) for every t ∈ R, i.e. A(y, t)λ, λ) ≥ α|λ|2, |A(y, t)λ| ≤ β|λ|, ∀ λ ∈ RN, a.e. in Y . (iii) there exists a function ω : R → R such that

ω is continuous, nondecreasing and ω(t) > 0 ∀ t > 0,
|A(y, t) − A(y, t1)| ≤ ω(|t − t1|) for a.e. y and t = t1,
for any r > 0, lim

s→0+

r

s

dt ω(t) = +∞.

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Position of the problem

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Then, we set Aε(x, t) = A x ε , t

,

for every (x, t) ∈ RN × R. Remark Assumptions (i)and(ii) are sufficient to prove the existence of a solution of problem. The additional condition (iii) allows to prove the uniqueness of the

solution. It has been introduced by M. Chipot (2009), in order to

prove the uniqueness of the above quasilinear elliptic equation with Dirichlet boundary conditions. It implies that ω(t) → 0 when t → 0. In particular, if A is Lipschitz continuous in t uniformly with respect to y, i.e., if there exists L > 0 s.t. |A(y, t) − A(y, t1)| ≤ L|t − t1|, for almost every y ∈ Y and for every t, t1 ∈ R, then the function ω(t) = Lt satisfies condition (iii) (as well as (i) and (ii)).

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Position of the problem

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Concerning the data in the nonlinear Robin condition Aε(x, uε)∇uε · n + εγτε(x)h(uε) = gε(x) on Γε

1,

we assume that

g is a Y -periodic function in L2(∂S);
τ is a positive Y -periodic function in L∞(∂S)

and, denoted M∂S(g) = 1 |∂S|

∂S

g(y) dσ

y, we set

τε(x) = τ x ε

,

gε(x) = g x

ε

if M∂S(g) = 0,

εg x

ε

if M∂S(g) = 0.

Remark This is the good scaling for gε in order or to have uniform and non trivial a priori estimates.

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Position of the problem

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Concerning the nonlinear term h(uε) we assume that              (i) h is an increasing function in C 1(R), with h(0) = 0; (ii) there exists a constant C > 0 and an exponent q with 1 ≤ q < ∞ if N = 2 and 1 ≤ q ≤ N N − 2 if N > 2, such that ∀s ∈ R,

h′(s)
≤ C(1 + |s|q−1).

Remarks ⋆ These assumptions have been introduced by D. Cioranescu, P. D. and

R. Zaki for the homogenization of the linear elliptic equation with

nonlinear Robin conditions. ⋆ In particular, they imply that sh(s) ≥ 0 and, for some positive constant C1, |h(s)| ≤ C1(1 + |s|q) ∀s ∈ R.

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Some physical motivations

For several composites the thermal conductivity depends in a nonlinear way on the temperature itself (see the Encyclopedia of Material Science and Engineering). For instance, for glass or wood, where the conductivity is nonlinearly increasing with the temperature, as well as ceramics, where it is decreasing, or aluminium and semi-conductors, where the dependence is not even monotone. Nonlinear Robin conditions appear in several physical situations such as climatization (see C. Timofte, Stud. Univ. Babes-Bolyai Math., 2007) or some chemical reactions (see C. Conca, J.I. Diaz, A. Linan and C. Timofte, Electronic Journal of Differential Equations, 2004).

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Position of the problem

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The variational formulation

For p ∈ [1, +∞), we define V p

ε =

φ ∈ W 1,p(Ω∗

ε) | φ = 0 on Γε

and

Vε = V 2

ε ,

which is a Banach space for the norm uV p

ε = ∇uLp(Ω∗ ε) ∀u ∈ V p

ε .

Then, the variational formulation of the problem is              Find uε ∈ Vε such that

Ω∗

ε

Aε(x, uε)∇uε∇v dx + εγ

Γε

1

τε(x)h(uε)v dσ

x

=

Ω∗

ε

fv dx +

Γε

1

gε(x)v dσ

x,

∀ v ∈ Vε.

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Existence and uniqueness of a solution

Existence and uniqueness of a solution of our problem has been proved in

B. Cabarrubias and P. Donato, Existence and Uniqueness for a

Quasilinear Elliptic Problem With Nonlinear Robin Conditions, to appear in Carpathian J. Math., (2) 2011. For the quasilinear problem with Dirichlet boundary conditions we refer to

M. Chipot, Elliptic Equations: An Introductory rse. Germany:

Birkhauser Verlag AG, 2009. For the quasilinear problem with linear Robin boundary conditions to

S. Bendib, Homog´

en´ eisation d’une classe de probl` emes non lin´ eaires avec des conditions de Fourier dans des ouverts perfor´ es, Th` ese, Institut National Polytechnique de Lorraine, 2004, and, for the linear equation with nonlinear Robin boundary conditions to

D. Cioranescu, P. Donato and R. Zaki, Asymptotic behavior of elliptic

problems in perforated domains with nonlinear boundary conditions, Asymptotic Analysis 53 (2007), 209-235.

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The Existence and uniqueness result

Let O be an open set of RN and suppose that its boundary ∂O is the union

f two disjoint closed surfaces, the exterior one Γ0 and the interior one Γ1.

Consider the problem      −div(B(x, u)∇u) = f in O, u = 0

n Γ0,

B(x, u)∇u · n + γ h(u) = g

n Γ1,

where n is the unit exterior normal to O, γ is a bounded positive function, f ∈ L2(O) and g ∈ L2(Γ1). Theorem Suppose that B and h satisfy the same assumptions as t A and h above (in O and without the periodicity). There exists a unique solution u in the space V , defined for every p ∈ [1, +∞[ by V p =

φ ∈ W 1,p(O)|φ = 0 on Γ0
,

and set V = V 2.

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Idea of the proof

The uniqueness is proved by similar arguments as those introduced by

Chipot for Dirichlet conditions. We skip here the proof, very technical.

We give here the proof of the existence, which as for Dirichlet

conditions, make use of the Schauder Theorem. In our case, the proof is more delicate and needs specific tools, due to the presence of both nonlinearity, that of the quasilinear term and that in the boundary conditions. The Schauder’s Fixed Point Theorem Let K be a closed convex set in a Banach space X. Let T be a continuous function from K to K such that T(K) is relatively compact. Then T has a fixed point in K.

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Two main preliminary results

Proposition 1 Let t ≥ 1 given by t ∈ [1, 2[ if N = 2 and q > 1, t = 2N q(N − 2) + 2

therwise.

Then, h acts from V to V t and the map u ∈ V → h(u) ∈ V t is bounded. Theorem 1 There exists r′ > 1 such that the map u ∈ V → h(u) ∈ Lr′(Γ1), is weakly continuous. That is, if {un}n∈N is a sequence in V such that un ⇀ u weakly in V , then, h(un) ⇀ h(u) weakly in Lr′(Γ1).

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Proof of the existence

Fix w ∈ L2(O) and consider the problem      Find u ∈ V such that ∀v ∈ V

O

B(x, w)∇u∇v dx +

Γ1

γh(u)v dσ =

O

fv dx +

Γ1

gv dσ. From the results from Cioranescu-D.-Zaki, this problem admits a unique

solution. We define the function T by

T : w ∈ L2(O) − → T(w) = u ∈ L2(O). Taking u as a test function we have α ∇u2

L2(O)

≤

O

B(x, w)∇u∇u dx +

Γ1

γh(u)u dσ =

O

fu dx +

Γ1

gu dσ ≤ (f L2(O) + c gL2(Γ1)) uH1(O) , since γh(u)u ≥ 0.

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This imples that there exists C > 0 such that uH1(O) = T(w)H1(O) ≤ C, for every w ∈ L2(O0). For this constant, define K as K =

v|v ∈ L2(O), vL2(O) ≤ C
,

which is a closed convex subset of L2(O). Let us show that T : K − → K satisfies the conditions of the Schauder Theorem. By construction, T(w)L2(O) ≤ C, ∀ w ∈ K, so that T(K) ⊂ K. Moreover, if {un}n∈N is a bounded sequence in K, then the sequence {T(un)} is also bounded in V . Consequently, from the Rellich Compactness Theorem, {T(un)} is relatively compact in L2(O), hence in K.

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It remains to show the continuity of T, which is the most delicate point. Let {wn}n∈N be a sequence in L2(O) such that wn → w strongly in L2(O) and set un = T(wn). The sequence {un} satisfies the equation

O

B(x, wn)∇un∇v dx =

O

fv dx +

Γ1

gv dσ −

Γ1

γh(un)v dσ ∀v ∈ V . Then, using the a priori estimates, we can extract a subsequence {unk}k∈N from {un} and a subsequence {wnk}k∈N from {wn} such that      (i) unk ⇀ u∗ weakly in V , (ii) unk → u∗ strongly in L2(O) and a.e. in O, (ii) wnk → w a.e. in O.

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To show that u∗ = T(w), we pass to the limit in the variational formulation above (written for the subsequence nk). We have B(x, wnk(x)) → B(x, w(x)) a.e. in O. Then, from the properties of B, tB(x, wnk(x))∇v →

tB(x, w(x))∇v

a.e. in O, (|tB(x, wnk)∇v| ≤ β|∇v| a.e. in O, for every k ∈ N and for every v ∈ V . Applying Lebesgue Dominated Convergence Theorem, we get

tB(x, wnk(x))∇v → tB(x, w(x))∇v

in [L2(O)]n. This, together with the above convergences, give lim

k→+∞

O

B(x, wnk)∇unk∇v dx = lim

k→+∞

O

tB(x, wnk)∇v∇unk dx

=

O

tB(x, w)∇v∇u∗ dx =

O

B(x, w)∇u∗∇v dx.

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For the boundary term, we use Theorem 1 to get h(unk) ⇀ h(u∗) weakly in Lr′(Γ1), for some r′ > 1. This gives lim

k→+∞

Γ1

γh(unk)v dσ

=
Γ1

γh(u∗)v dσ. Consequently,

O

B(x, w)∇u∗∇v dx =

O

fv dx +

Γ1

gv dσ −

Γ1

γh(u∗)v dσ, for every v ∈ V . By uniqueness, we have u∗ = T(u) and the continuity of T. Hence, from the Schauder Theorem, T has a fixed point in L2(O), which his gives the existence of a solution of our problem.

Patrizia Donato (University of Rouen)

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Statement of the homogenization results

Theorem 2 Under assumptions above, let uε be the unique solution of the problem, with γ ≥ 1 and t > 1 given by

t ∈ (1, 2),

if N = 2 and q > 1,

t =

2N q(N − 2) + 2,

therwise.

Then, there exists (u0, u) in H1

0(Ω) × L2(Ω; H1 per(Y ∗)) with MY ∗(ˆ

u) = 0, such that          (i) T ∗

ε (uε) → u0

strongly in L2(Ω; H1(Y ∗)), (ii) T ∗

ε (∇uε) ⇀ ∇u0 + ∇y

u weakly in L2(Ω × Y ∗), (iii) T b

ε (h(uε)) → h(u0)

strongly in Lt(Ω; W 1− 1

t ,t(∂S)). Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

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If γ = 1 the pair (u0, u) is the unique solution in H1

0(Ω) × L2(Ω; H1 per(Y ∗))

with MY ∗(ˆ u) = 0, of the limit equation                          ∀φ ∈ H1

0(Ω), ∀Ψ ∈ L2(Ω; H1 per(Y ∗))

Ω×Y ∗ A(y, u0)(∇u0 + ∇y

u)(∇φ(x) + ∇yΨ(x, y)) dx dy +|∂S|M∂S(τ)

Ω

h(u0)φ dx = |Y ∗|

Ω

f φ dx + |∂S|M∂S(g)

Ω

φ dx.

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If γ > 1 the pair (u0, u) is the unique solution in H1

0(Ω) × L2(Ω; H1 per(Y ∗))

with MY ∗(ˆ u) = 0, of the limit equation                ∀φ ∈ H1

0(Ω), ∀Ψ ∈ L2(Ω; H1 per(Y ∗))

Ω×Y ∗ A(y, u0)(∇u0 + ∇y

u)(∇φ(x) + ∇yΨ(x, y)) dx dy = |Y ∗|

Ω

f φ dx + |∂S|M∂S(g)

Ω

φ dx.

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Corollary 1 Under the assumptions and notations of Theorem 2, let γ ≥ 1 and denote the extension by 0 to Ω. Then,

uε ⇀ |Y ∗|

|Y | u0 weakly in L2(Ω), where denotes the extension by 0 to Ω. If γ = 1, the function u0 is the unique solution of the limit problem      − div(A0(u)∇u0) + |∂S| |Y | M∂S(τ)h(u0) = |Y ∗| |Y | f + |∂S| |Y | M∂S(g) in Ω, u0 = 0

n ∂Ω

and, if γ > 1, the function u0 is the unique solution of the problem      − div(A0(u)∇u0) = |Y ∗| |Y | f + |∂S| |Y | M∂S(g) in Ω, u0 = 0

n ∂Ω.

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The homogenized matrix field A0(t) is given by A0(t)λ = 1 |Y |

Y ∗ A(y, t)∇wλ(t, y)dy,

∀λ ∈ RN, where wλ(y, t) = −χλ(y, t) + λ · y a.e. in Y ∗ and χλ(·, t) is, for every t, the solution of the problem            − div(A(·, t)∇χλ(·, t)) = − div(A(·, t)λ) in Y ∗, A(·, t)∇(χλ(·, t) − λ · y) · n = 0

n ∂S,

χλ(·, t) Y − periodic, MY ∗(χλ(·, t)) = 0. Remark If condition (iii) on A is not assumed, all the convergences still hold for a subsequence of uε and u0 is a (not necessarily unique) solution of the limit problem.

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Some references

The above homogeneization results are contained in

B. Cabarrubias and P. Donato, Homogenization of a quasilinear elliptic

problem with nonlinear Robin boundary conditions, Applicable Analysis, (2011) to appear. For the homogenization of a periodic quasilinear elliptic problem with Lipschitz continuous coefficients in a fixed domain we refer to

M.Artola and G. Duvaut, Un r´

esultat d’homog´ en´ eisation pour une classe de probl` emes de diffusion non lin´ eaires stationnaires, Annales Facult´ e des Sciences Toulouse, Vol. IV (1982), 1-27. For generalizations in the framework of the H-convergence see

L. Boccardo and F. Murat, Homog´

en´ eisation de probl` emes quasi lin´ eaires, Proc. Problemi-limite dell’analisi funzionale, Bressanone 1981, Pitagora ed. Bologna (1982), 13-53.

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For periodically perforated domains, linear equation with linear Robin condition has been studied in

D. Cioranescu and P. Donato, Homog´

en´ esation du probl` eme de Neumann non homog` ene dans des ouverts perfor´ es, Asymptotic Anal. 1 (1988), 115-138, and with nonlinear Robin conditions in

D. Cioranescu, P. Donato and R. Zaki, Asymptotic behavior of elliptic

problems in perforated domains with nonlinear boundary conditions, Asymptotic Analysis 53 (2007), 209-235. The case of quasilinear elliptic equation with Lipschitz continuous coefficients and linear Robin conditions has been studied in

S. Bendib and R.L. Tcheugou´

e T´ ebou, Homog´ en´ eisation d’une classe de probl` emes non lin´ eaires dans des domaines perfor´ es, C.R. Acad. Sci. Paris,

t. 328, Serie 1 (1999), 1145-1149.
S. Bendib, Homog´

en´ eisation d’une classe de probl` emes non lin´ eaires avec des conditions de Fourier dans des ouverts perfor´ es, Th` ese, Institut National Polytechnique de Lorraine, 2004.

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Properties of the quasilinear homogenized matrix

One of the main difficulties is to show that also the homogenized problem has a unique solution. We prove that the (quasilinear) homogenized matrix field satisfies the same kind of assumptions as the original problem. To do that, we use a pointwise estimate on the difference of two homogenized matrices for the linear problem perforated domains in terms of the difference of the original ones.

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The quasilinear homogenized matrix

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The homogenized matrix of the linear problem For any matrix field B ∈ M(α, β, Y ), let B0 be the homogenized matrix for perforated domains introduced by D. Cioranescu and J. Saint Jean Paulin (1979), given by B0λ = 1 |Y |

Y ∗ B(y)∇wB

λ (y)dy,

∀λ ∈ RN, where wB

λ (y) = −χB λ (y) + λ · y

for almost every y ∈ Y ∗ and χB

λ is the solution of the problem

           − div(B∇χB

λ ) = − div(Bλ)

in Y ∗, B∇(χB

λ − λ · y) · n = 0

n ∂S,

χB

λ

Y − periodic, MY ∗(χB

λ ) = 0.

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

The quasilinear homogenized matrix

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trix

Set χB

λε(x) = εχB λ

x ε

,

wB

λε(x) = εwB λ

x ε

(= −χB

λε(x) + λ · x),

for almost every x ∈ Ω∗

ε.

Recall that   

χB

λε → 0

strongly in L2(Ω), Bε ∇wB

λε ⇀ B0λ

weakly in [L2(Ω)]N, where denotes the extension by 0 to Ω, and, by a change of scale,

Ω∗

ε

Bε∇wB

λε∇v dx = 0

∀v ∈ H1

0(Ω).

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

The quasilinear homogenized matrix

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Proposition 2 (a comparison result) Let B and D be in M(α, β, Y ) and B0 and D0 be the corresponding homogenized matrices. Then, there exists C1 > 0 depending on α, β, Y and S such that |B0 − D0| ≤ C1 B − DL∞(Y ) . Remark Similar results have been proved in a nonperiodic framework for a fixed domain in

Colombini-Spagnolo (1977) for the G-convergence,
Boccardo-Murat (1982) for the H-convergence.

For perforated domains and the H-convergence of the linearized elasticity system, see Donato-Haddadou (2006). ⋆ We give here the proof for a periodically perforated domain, which is simpler and follows some arguments used for a fixed domain in Cioranescu-Donato (1999).

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

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Lemma 1 There exists q > 2 depending only on α, β, Y and S such that the function wB

λ belongs to W 1,q(Y ∗). Moreover, there exists a constant

C depending only on α, β, Y , S and p such that

∇wB

λ

Lp(Y ∗) ≤ C|λ|,

for every p ∈ [2, q]. Also, lim

ε→0

∇wB

λε

Lp(Ω)

≤ |Ω|

1 p

|Y |

1 p

C|λ|. Proof From the Meyers estimate (see also Briane-Damlamian-Donato (1998) for the Neumann conditions), ∃ q = q(α, β, Y , S) > 2 such that χB

λ is in

W 1,q(Y ∗). Moreover, for same c = (α, β, Y , S, p) we have

∇χB

λ

Lp(Y ∗) ≤ c BλLp(Ω) ≤ cβ|λ|,

for every p ∈ [2, q].

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

The quasilinear homogenized matrix

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trix

Hence,

∇wB

λ

Lp(Y ∗) =
∇χB

λ − λ

Lp(Y ∗) ≤ (cβ + 1)|λ|,

which gives the first estimate for C = cβ + 1. Then, classical results on functions of the form f (x/ε) imply lim

ε→0

∇wB

λε

p

Lp(Ω)

= lim

ε→0

Ω∗

ε

∇ywB

λ

x ε

p

dx = |Ω| |Y |

∇wB

λ

p

Lp(Y ∗) ≤ |Ω|

|Y |C p|λ|p, which ends the proof.

Patrizia Donato (University of Rouen)

Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

The quasilinear homogenized matrix

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Proof of Proposition 2 For λ, µ ∈ RN, consider the functions χB

λε, wB λε, χ

tD

µε and w

tD

µε defined for

the matrix fields B and tD ( the transposed matrix field of D). Let us show that Bε ∇wB

λε

∇w

tD

µε ⇀ B0λµ

in D′(Ω). We have, for any φ ∈ D(Ω), lim

ε→0

Ω

Bε ∇wB

λε

∇w

tD

µε φ dx

= lim

ε→0

Ω∗

ε

Bε ∇wB

λε ∇(w

tD

µεφ) dx −

Ω∗

ε

Bε ∇wB

λε w

tD

µε ∇φ dx

= lim

ε→0

Ω

Bε ∇wB

λε (

χ

tD

µε − µ · x) ∇φ dx = −

Ω

B0λ(µ · x) ∇φ dx =

Ω

B0λµφ dx.

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

The quasilinear homogenized matrix

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trix

A similar computation gives, lim

ε→0

Ω

tDε

∇w

tD

µε

∇wB

λε φ dx =

Ω

tD0µλ φ dx,

which implies

tDε

∇w

tD

µε

∇wB

λε ⇀ tD0µλ

in D′(Ω), so that Dε ∇wB

λε

∇w

tD

µε ⇀ D0λµ

in D′(Ω). Then, (Bε − Dε) ∇wB

λε

∇w

tD

µε ⇀ (B0 − D0)λµ

in D′(Ω), for every µ, λ ∈ RN. On the other hand, from Lemma 1, (Bε − Dε) ∇wB

λε

∇w

tD

µε is bounded in

L1+η(Ω). Hence, (Bε − Dε) ∇wB

λε

∇w

tD

µε ⇀ (B0 − D0)λµ

weakly in L1(Ω), for every µ, λ ∈ RN.

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

The quasilinear homogenized matrix

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trix

This gives, using again Lemma 1, |Ω||(B0 − D0)λµ| =

(B0 − D0)λµ
L1(Ω)

≤ lim inf

ε→0

(Bε − Dε)

∇wB

λε

∇w

tD

µε

L1(Ω)

≤ B − DL∞(Y ) lim

ε→0

∇wB

λε

L2(Ω)

lim

ε→0

∇w

tD

µε

L2(Ω)

≤ |Ω| |Y |C 2 B − DL∞(Y ) |λ||µ|, since B0 and D0 are constant and Bε − DεL∞(Ω) = B − DL∞(Y ). Hence, |(B0 − D0)λµ| ≤ C1 B − DL∞(Y ) |λ||µ|, where C1 depends on α, β, Y and S. Choosing λ = ei, µ = ej, where (ei)N

i=1 is the canonical basis of RN, gives the result.

Patrizia Donato (University of Rouen)

Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

The quasilinear homogenized matrix

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lem

Uniqueness of the homogenized problem

The following result state that the homogenized operator inherits the modulus of continuity of the initial problem: Theorem 3 The matrix A0 satisfies the following properties: (1) A0 is continuous on R and A0(t) ∈ M

α, β2/α, Ω
, for every t ∈ R;

(2) there exists a positive constant C, depending only on α, β, Y and S such that |A0(t) − A0(t1)| ≤ C ω(|t − t1|), for every t = t1, where ω is the continuity modulus of A. Proof The second property in (1) follows from classical results. The others are straightforward consequences of the comparison result, written for the matrix fields B = A(·, t) and D = A(·, t1).

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011 • Uniqueness of the homogenized problem40/59

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lem

Corollary 2 Under the above assumptions, for every f1 ∈ L2(Ω) and α1 ∈ R∗

+, the

problem

− div (A0(u)∇u) + α1h(u) = f1

in Ω, u = 0

n ∂Ω,

admits a unique solution u ∈ H1

0(Ω).

Proof

If h ≡ 0, the result follows from the book of M. Chipot, in view of Theorem 3. In the case h ≡ 0, the proof can be adapted following the argument given in Cabarrubias-Donato for the case of nonlinear boundary condition.

Patrizia Donato (University of Rouen)

Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011 • Uniqueness of the homogenized problem41/59

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method

A short review of the unfolding method

Originally introduced in

D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and

homogenization, C.R. Acad. Sci. Paris, Ser. I 335 (2002), 99-104. For a general presentation and detailed proofs see

D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding and

homogenization, SIAM J. Math. Anal., Vol. 40, No. 4 (2008), 1585-1620.

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

About the unfolding method

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method

It has extended to perforated domains in

D. Cioranescu, P. Donato and R. Zaki, The Periodic Unfolding and

Robin problems in perforated domains, C.R.A.S. Paris, S´ erie 1, 342 (2006), 467-474.

D. Cioranescu, P. Donato and R. Zaki, The Periodic Unfolding Method

in Perforated Domains, Portugal Math 63(4) (2006), 476-496.

D. Cioranescu, P. Donato and R. Zaki, Asymptotic behavior of elliptic

problems in perforated domains with nonlinear boundary conditions, Asymptotic Analysis 53 (2007), 209-235. For more general situations and a comprehensive presentation, see

D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki,The

Periodic Unfolding Method in Domains With Holes, submitted for publication.

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

About the unfolding method

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method

⋆ We recall here only the definitions and the properties for perforated domains needed in the following. Let [z]Y =

N

j=1

kjbj be the unique integer combination of periods such that for any z ∈ RN, z − [z]Y is in Y and set {z}Y = z − [z]Y . In particular, for positive ε, x = ε x ε

Y +

x ε

Y
for all

x ∈ RN. We also denote by MY ∗ the mean value of an integrable function on Y ∗, given by MY ∗(Φ) = 1 |Y ∗|

Y ∗ Φ(y)dy,

∀Φ ∈ L1(Y ∗).

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

About the unfolding method

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method

Definition 1 For any Lebesgue-measurable function φ on Ω∗

ε, the unfolding operator

T ∗

ε

is defined as T ∗

ε (φ)(x, y) =

φ
ε

x ε

Y + εy
,

a.e. for (x, y) ∈ Ωε × Y ∗ a.e. for (x, y) ∈ Λε × Y ∗. Definition 2 For any function ϕ Lebesgue-measurable on ∂ Ω∗

ε ∩ ∂Sε, the boundary

unfolding operator T b

ε

is defined by T b

ε (ϕ)(x, y) =

ϕ
ε

x ε

Y + εy
,

a.e. for (x, y) ∈ Ωε × ∂S, a.e. for (x, y) ∈ Λε × ∂S. Remark T ∗

ε is linear and continuous from Lp(Ω∗ ε) to Lp(Ω × Y ∗).

T b

ε is a linear operator.

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

About the unfolding method

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method

Proposition 3 Let p ∈ [1, +∞). Then, T ∗

ε (φψ) = T ∗ ε (φ)T ∗ ε (ψ) for every φ, ψ ∈ Lp(Ω∗ ε).

For w ∈ Lp(Ω), T ∗

ε (w) → w

strongly in Lp(Ω × Y ∗). For all φ ∈ L1(Ω∗

ε) one has

Ω∗

ε

φ(x) dx =

Ω∗

ε

φ(x) dx−

Λ∗

ε

φ(x) dx = 1 |Y |

Ω×Y ∗ T ∗

ε (φ)(x, y) dx dy.

∇yT ∗

ε (φ)(x, y) = εT ∗ ε (∇xφ)(x, y) for every (x, y) ∈ RN × Y ∗.

If φε → φ strongly in Lp(Ω), then T ∗

ε (φε) → φ

strongly in Lp(Ω × Y ∗). Let φ ∈ Lp (Y ∗) be a Y −periodic function and set φε (x) = φ x ε

.

Then, T ∗

ε (φε) (x, y) = ϕ(y) a.e. in Ω × Y ∗.

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

About the unfolding method

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method

Proposition 4 Let p ∈ (1, +∞). Then, T b

ε (φψ) = T b ε (φ)T b ε (ψ) for every φ, ψ ∈ Lp(∂Sε).

Let φ ∈ Lp(∂S) be a Y -periodic function. Set φε(x) = φ x

ε

. Then,

T b

ε (φε)(x, y) = φ(y) a.e. in Ω × ∂S.

For all φ ∈ L1(∂Sε), the integration formula is given by

Γε

1

φ(x) dσ

x =

1 ε|Y |

Ω×∂S

T b

ε (φ)(x, y) dx dσ y.

Let φ ∈ H1(Ω∗

ε). Then

T b

ε (φ) → φ

strongly in Lp(Ω × ∂S).

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

About the unfolding method

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method

An important tool for the homogenization of our problem is the following proposition: Proposition 5 Let g ∈ L2(∂S), Y -periodic, and let gε be defined as above. Then, for every φ ∈ H1(Ω∗

ε) such that φ = 0 on Γε 0, one has

Γε

1

g(x ε )φ(x) dσ

x

≤ C

ε (|M∂S(g)| + ε) ∇φ[L2(Ω∗

ε)]N .

Furthermore, if φ ∈ D(Ω), lim

ε→0

Γε

1

gε(x)φ(x) dσ

x = |∂S|

|Y | M∂S(g)

Ω

φ(x) dx.

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

About the unfolding method

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Some preliminary results

In order to prove the homogenization results, we give the following propositions: Proposition 6 Under the above assumptions, let uε be the unique solution of our

problem. Then, there exists C independent of ε such that

uεVε ≤ C. Moreover, there exists a subsequence, still denoted by ε, u0 ∈ H1

0(Ω) and

u ∈ L2(Ω; H1

per(Y ∗)) such that

(i) T ∗

ε (uε) → u0

strongly in L2(Ω; H1(Y ∗)), (ii) T ∗

ε (∇uε) ⇀ ∇u0 + ∇y

u weakly in L2(Ω × Y ∗), (iii) T ∗

ε (Aε (x, uε)) → A(y, u0)

a.e. in Ω × Y ∗.

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

Some preliminary results

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Proof The a priori estimate follows from Proposition 5, while convergences (i) and (ii) follows from known results of the unfolding method. Let us prove (iii). By definition, T ∗

ε

Aε

(x, uε(x)) = A x ε

Y + y, uε
ε

x ε

Y + εy
= A

x ε

Y + y, T ∗

ε (uε)(x, y)

= A (y, T ∗

ε (uε)(x, y)) ,

a.e. in Ωε × Y ∗ and T ∗

ε

Aε

(x, uε(x)) = 0 a.e. in Λε × Y ∗. On the other hand, from (i), there exists a subsequence (still denoted ε) such that T ∗

ε (uε) converges to u0 a.e. in Ω × Y ∗.

This, together with the Carath´ eodory assumption on A, yields A(y, T ∗

ε (uε)(x, y)) → A(y, u0)

a.e. in Ω × Y ∗, which gives the result.

Patrizia Donato (University of Rouen)

Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

Some preliminary results

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Theorem 4 Under the assumptions and notations of Proposition 6, let t > 1 as in Theorem 2. Then, T b

ε (h(uε)) → h(u0)

strongly in Lt(Ω; W 1− 1

t ,t(∂S)).

Proof We follow some arguments used in Cioranescu-D.-Zaki for N > 2. From Proposition 6 in particular, T ∗

ε (uε) → u0

strongly in L2(Ω × Y ∗). Moreover, T ∗

ε (uε) is bounded in Ls(Ω × Y ∗) for

s = 2∗

if N > 2, s ∈ (q, +∞) if N = 2. Indeed, from (4) of Proposition 4 and Proposition 6, T ∗

ε (uε)Ls(Ω×Y ∗) ≤ C uεLs(Ω) ≤ C1 uεVε ≤ C2.

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

Some preliminary results

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Hence, by interpolation that T ∗

ε (uε) → u0

strongly in Lr(Ω × Y ∗), for every r ∈ [2, s). Choosing r = q if N = 2 and 2∗

2 ≤ r < 2∗ if N > 2

gives r ≥ q. Thus, classical results from Krasnoselskii give h(T ∗

ε (uε)) = T ∗ ε (h(uε)) → h(u0)

strongly in L

r q (Ω × Y ∗).

Also, from Proposition 1, h(uε)V t

ε is bounded for t defined above.

Then, a known unfolding result imply that there exists h∗ ∈ V t

ε s.t., up to

a subsequence, T ∗

ε (h(uε)) → h∗

strongly in Lt(Ω; W 1,t(Y ∗)), which, together with the above convergence, gives h(u0) = h∗. Consequently, T ∗

ε (h(uε)) → h(u0)

strongly in Lt(Ω; W 1,t(Y ∗)). Due to the linearity of the trace operator, this gives the result.

Patrizia Donato (University of Rouen)

Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

Some preliminary results

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Proof of Theorem 2

Proposition 6 and Theorem 4 give the convergences for a subsequence (still denoted ε). We only identify here the problem satisfied by (u0, ˆ u). The uniqueness of the solution (u0, ˆ u) is given proof of Corollary 2 and implies that the convergences hold for the whole sequence. Consider first the case γ = 1. Choosing φ ∈ D(Ω) as a test function in the problem gives       

Ω∗

ε

Aε(x, uε)∇uε∇φ dx + εγ

Γε

1

τε(x)h(uε)φ dσ

x

=

Ω∗

ε

f φ dx +

Γε

1

gε(x)φ dσ

x.

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

Proofs of the homogenization results 53/59

SLIDE 54

sults

By using the unfolding method, from Proposition 3 we obtain lim

ε→0

Ω∗

ε

Aε(x, uε)∇uε∇φ dx = lim

ε→0

1 |Y |

Ω×Y ∗ T ∗

ε (Aε(x, uε)) T ∗ ε (∇uε)T ∗ ε (∇φ) dx dy

= lim

ε→0

1 |Y |

Ω×Y ∗ T ∗

ε

tAε(x, uε)

T ∗

ε (∇φ)T ∗ ε (∇uε) dx dy,

since, for ε sufficiently small, φ = 0 in Λ∗

ε.

Observe now that, in view of of Proposition 3 and Proposition 6, using the assumptions on A we can apply the Lebesgue Dominated Convergence Theorem to derive that T ∗

ε

tAε(x, uε)

T ∗

ε (∇φ) → A(y, u0)∇φ

strongly in L2(Ω × Y ∗), This, together with Proposition 6 allows to pass to the limit to obtain lim

ε→0

Ω∗

ε

Aε(x, uε)∇uε∇φ dx = 1 |Y |

Ω×Y ∗ A(y, u0)(∇u0 + ∇y

u)∇φ dx dy.

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

Proofs of the homogenization results 54/59

SLIDE 55

sults

Also, using Proposition 4, ε

Γε

1

τε(x)h(uε)φ dσ

x =

1 |Y |

Ω×∂S

τ(y)T b

ε (h(uε))T b ε (φ) dx dσ y.

On the other hand, from Theorem 4 one has T b

ε (h(uε)) → h(u0)

strongly in Lt(Ω × ∂S), where t > 1 and from Proposition 4 T b

ε (φ) → φ

strongly in Lt′(Ω × ∂S), where t′ is the conjugate of t. Thus, since φ and u0 are functions of x only, one has lim

ε→0 ε

Γε

1

τε(x)h(uε)φ dσ

x

= 1 |Y |

Ω×∂S

τ(y)h(u0)φ dx dσ

y

= 1 |Y |

∂S

τ(y) dσ

y Ω

h(u0)φ dx

.

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

Proofs of the homogenization results 55/59

SLIDE 56

sults

Finally, using again Proposition 3, lim

ε→0

Ω∗

ε

f φ dx = lim

ε→0

1 |Y |

Ω×Y ∗ T ∗

ε (f )T ∗ ε (φ) dx dy =

1 |Y |

Ω×Y ∗ f φ dx dy.

Then, passing to the limit since φ and f are functions of x only, one has

Ω×Y ∗ A(y, u0)(∇u0

+ ∇y u)∇φ dx dy + |∂S|M∂S(τ)

Ω

h(u0)φ dx = |Y ∗|

Ω

f φ dx + |∂S|M∂S(g)

Ω

φ dx, which is still valid for every φ ∈ H1

0(Ω) by density.

Next, for φ ∈ D(Ω) and ξ ∈ H1

per(Y ∗) take as a test vε = εφ(x)ξ

x

ε

.

Observe that T ∗

ε (vε) = εT ∗ ε (φ)ξ, and ∇vε = ε∇φξ

·

ε

+ φ∇yξ

·

ε

.

Then, from Proposition 3, T ∗

ε (vε) → 0

strongly in L2(Ω × Y ∗), T ∗

ε (∇vε) → φ∇yξ

strongly in L2(Ω × Y ∗).

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

Proofs of the homogenization results 56/59

SLIDE 57

sults

By unfolding, from Proposition 4, one has lim

ε→0 ε

Γε

1

τε(x)h(uε)vε dσ

x = lim ε→0

1 |Y |

Ω×∂S

τ(y)T b

ε (h(uε))T b ε (vε) dx dσ y = 0

and, using Proposition 3, lim

ε→0

Ω∗

ε

f vε dx = lim

ε→0

1 |Y |

Ω×Y ∗ T ∗

ε (f )T ∗ ε (vε) dx dy = 0.

Consequently, arguing as before, one has lim

ε→0

Ω∗

ε

Aε(x, uε)∇uε∇vε dx = lim

ε→0

1 |Y |

Ω×Y ∗ T ∗

ε (Aε(x, uε))T ∗ ε (∇uε)T ∗ ε (∇vε) dx dy

= 1 |Y |

Ω×Y ∗ A(y, u0)(∇u0 + ∇y

u)φ(x)∇yξ(y) dx dy = 0.

Patrizia Donato (University of Rouen) Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

Proofs of the homogenization results 57/59

SLIDE 58

sults

By density, for every Ψ ∈ L2(Ω; H1

per(Y ∗)),

1 |Y |

Ω×Y ∗ A(y, u0)(∇u0 + ∇y

u)∇yΨ(x, y) dx dy = 0. Summing this equality with the equality

Ω×Y ∗ A(y, u0)(∇u0

+ ∇y u)∇φ dx dy + |∂S|M∂S(τ)

Ω

h(u0)φ dx = |Y ∗|

Ω

f φ dx + |∂S|M∂S(g)

Ω

φ dx,

btained before, gives the limit equation in Theorem 2.

The proof for γ > 1 follows by the same computation but, in this case, the nonlinear boundary term goes to zero at the limit.

Patrizia Donato (University of Rouen)

Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

Proofs of the homogenization results 58/59

SLIDE 59

sults

Proof of Corollary 1

Suppose γ ≥ 1. Choosing φ = 0 in the limit equation obtained in Theorem 2, gives here ˆ u(x, y) = −

N

i=1

χei(y, u0(x)) ∂u0 ∂xi (x) + κ(x) for some κ ∈ L2(Ω), where (ei) is the canonical basis of RN. Moreover, the condition MY ∗(ˆ u) = 0 gives k ≡ 0. A standard computation shows that u0 is solution in Ω of the claimed limit problem. On the other hand, Corollary 2 provides the uniqueness of u0, which gives the uniqueness of ∇ˆ u under the condition MY ∗(ˆ u) = 0. This proves that all the convergences hold for the whole sequences in Theorem 2. The convergence in Corollary 1 is then straightforward.

Patrizia Donato (University of Rouen)

Patrizia Donato (University of Rouen) Homogenization Symposium, Shanghai 2011

Proofs of the homogenization results 59/59