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DOI:10.1051/m2an/2012030 www.esaim-m2an.org

CORRECTORS AND FIELD FLUCTUATIONS FOR THE

p

(

x

)-LAPLACIAN WITH ROUGH EXPONENTS: THE SUBLINEAR GROWTH CASE

Silvia Jimenez

1

Abstract.A corrector theory for the strong approximation of gradient fields inside periodic composites made from two materials with different power law behavior is provided. Each material component has a distinctly different exponent appearing in the constitutive law relating gradient to flux. The correctors are used to develop bounds on the local singularity strength for gradient fields inside micro-structured media. The bounds are multi-scale in nature and can be used to measure the amplification of applied macroscopic fields by the microstructure. The results in this paper are developed for materials having power law exponents strictly between1 and zero.

Mathematics Subject Classification. 35J66, 35A15, 35B40, 74Q05.

Received September 11, 2011. Revised April 13, 2012.

Published online January 11, 2013.

1. Introduction

In this paper, we develop a corrector theory for the strong approximation of gradient fields inside periodic composites made from two materials with different power law behavior. Here the flux is related to the gradient

∇uby the power law|∇u|r∇u. Each material component has a distinctly different exponentrappearing in the constitutive law relating gradient to flux. The correctors are used to develop bounds on the local singularity strength for gradient fields inside micro-structured media. The bounds are multi-scale in nature and can be used to measure the amplification of applied macroscopic fields by the microstructure. The novelty of the work presented in this paper is that it is carried out for materials having power law exponentsrstrictly between1 and zero. Previous work [15], addresses power law materials with power law exponentsr greater than or equal to zero. For that case strong approximations to the gradient fields are developed and lower bounds on theLq norms (q2) of the gradient fields inside each material are provided. The bounds and strong approximations are expressed in terms of the correctors presented in Theorem 2.6 of [15].

The corrector theory for the linear case can be found in [20]. The earlier work of [9] provides the corrector theory for homogenization of monotone operators that in our case applies to composite materials made from con- stituents having the same power-law growth but with rough coefficientsσ(x). More recently, the homogenization ofp(x)-Laplacian boundary value problems for smooth exponential functions p(x) uniformly converging to a

Keywords and phrases.Correctors, field concentrations, dispersed media, homogenization, layered media,p-Laplacian, periodic domain, power-law materials, young measures.

1 Dept. of Mathematical Sciences, Worcester Polytechnic Institute 100 Institute Road, Worcester, 01609-2280 MA, USA.

silviajimenez@wpi.edu

Article published by EDP Sciences c EDP Sciences, SMAI 2013

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limit functionp0(x) has been studied in [1]. The convergence of the family of solutions for these homogenization problems is given in the topology ofLp0(·)(Ω).

Here we assume that the geometry of the composite is periodic and is specified by the indicator function of the sets occupied by each of the materials. The indicator function of material 1 and 2 are denoted byχ1 and χ2, where χ1(y) = 1 in material 1 and is zero outside and χ2(y) = 1−χ1(y). The constitutive law for the heterogeneous medium is described byA:Rn×Rn Rn,

A(y, ξ) =σ(y)|ξ|p(y)−2ξ, (1.1)

with σ(y) = χ1(y)σ1+χ2(y)σ2 and 0< σ1, σ2 <∞; with p(y) =χ1(y)p1+χ2(y)p2 and 1 < p1 ≤p2 <2 or 1 < p1 2 p2; and with both, σ(y) and p(y), periodic in y, with unit period cell Y = (0,1)n. This constitutive model occurs in several mathematical models of physical processes including nonlinear di- electrics [11,12,18,27,28], fluid flow (electrorheological fluids) [2,4,25], glaciology [13], image restoration [17], and in the theory of deformation plasticity under longitudinal shear (anti-plane strain deformation) [3,14,23,24,26].

In this paper, we study the problem of periodic homogenization associated with the solutions u to the problems

−div Ax

,∇u

=f onΩ,u∈W01,p1(Ω), (1.2)

whereΩis a bounded open subset ofRn,f ∈W−1,q2(Ω), and 1/p1+ 1/q2= 1. The differential operator on the left-hand side of (1.2) is thep(x)-Laplacian. All solutions are understood in the usual weak sense [30].

It was shown in Chapter 15 of [30] that {u}>0 converges weakly in W1,p1(Ω) to the solution u of the homogenizedproblem

div (b(∇u)) =f onΩ,u∈W01,p1(Ω), (1.3) where the monotone map b : Rn Rn (independent of f and Ω) can be obtained by solving an auxiliary problem for the operator (1.2) on a periodicity cell.

The idea of homogenization is intimately related to theΓ-convergence of a suitable family of energy functionals Ias0 [30]. Here the connection is natural in that the family of boundary value problems (1.2) correspond to the Euler equations of the associated energy functionals I and the solutions u are their minimizers. The homogenized solution is precisely the minimizer of theΓ-limit of the sequence{I}>0. The connections between Γ limits and homogenization for the power-law materials studied here can be found in Chapter 15 of [30]. The explicit formula for theΓ-limit of the associated energy functionals for layered materials was obtained recently in [22].

The homogenization result found in Chapter 5 of [30] shows that theaverage of the error incurred in approx- imating{∇u}>0in terms of∇u, whereuis the solution of (1.3) decays to 0. Then again, the presence of large local fields either electric or mechanical often precede the onset of material failure (see, [16]). The goal of our analysis is to develop tools for quantifying the effect of load transfer between length scales inside heterogeneous media. To this end, we present a new corrector result that approximates, inside each phase,∇uup to an error that converges to zero strongly in the norm (see Sect.2.2.1).

The corrector result is then used to develop new tools that provide lower bounds on the local gradient field intensity inside micro-structured media. The bounds are expressed in terms of the Lq-norms of gradients of the solutions of the local corrector problems. These results provide a lower bound on the amplification of the macroscopic gradient field by the microstructure see, Section 2.2.2. These bounds provide a rigorous way to assess the effect of field concentrations generated by the microgeometry without having to compute the actual solutionu. In [19], similar lower bounds were established for field concentrations for mixtures of linear electrical conductors in the context of two scale convergence.

In this paper, the corrector results are presented for layered materials (Fig.2) and for dispersions of inclusions embedded inside a host medium (Fig. 1). For the dispersed microstructures the included material is taken to have the lower power-law exponent than that of the host phase. The reason we use dispersed and layered microstructures is that in both cases we are able to show that the homogenized solution lies in W01,p2(Ω),

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F Y

Figure 1. Unit cell: dispersed microstructure.

R

2

R

1

R

2

Γ

Figure 2.Unit cell: layered material.

see Theorem 2.5. Possible extensions of this work include the study of other microstructures for which this higher order integrability condition of the homogenized solution is satisfied. The higher order integrability is used to provide an algorithm for building correctors and construct a sequence of strong approximations to the gradients inside each material, see Theorem 2.7. When the host phase has a lower power-law exponent than the included phase, one can only conclude that the homogenized solution lies in W01,p1(Ω) and the techniques developed here do not apply.

The presentation of the paper is organized as follows. In Section2, we state the problem and formulate the main results. Section 3contains technical lemmas and integral inequalities for the correctors used to prove the main results. Section 4 contains the proof of the main results. The Appendix contains all proofs of lemmas stated in Section3and some remarks related to the proof of Theorem2.7found in Section4.

2. Statement of the problem and main results

2.1. Notation

In this paper we consider two nonlinear power-law materials periodically distributed inside a domainΩ⊂Rn. The periodic mixture is described as follows. We introduce the unit period cellY = (0,1)nof the microstructure.

LetFbe an open subset ofY of material 1, with smooth boundary∂F, such thatF ⊂Y. The functionχ1(y) = 1 insideF and 0 outside andχ2(y) = 1−χ1(y). We extendχ1(y) andχ2(y) by periodicity toRnand the-periodic mixture insideΩis described by the oscillatory characteristic functionsχ1(x) =χ1(x/) andχ2(x) =χ2(x/).

Here we will consider the case whereF is given by a simply connected inclusion embedded inside a host material (see Fig.1). A distribution of such inclusions is commonly referred to as a periodic dispersion of inclusions.

We also consider layered materials. For this case the representative unit cell consists of a layer of material 1, denoted byR1, sandwiched between layers of material 2, denoted byR2. The interior boundary ofR1is denoted byΓ (see Fig.2). Hereχ1(y) = 1 fory∈R1and 0 inR2, andχ2(y) = 1−χ1(y).

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We denote by θ1 =

Y χ1(y)dy and θ2 = 1−θ1 the volume fractions of phase 1 and phase 2 inside the composite.

On the unit cellY, the constitutive law for the nonlinear material is given by (1.1) with exponentsp1 and p2satisfying 1< p1≤p22 or 1< p12≤p2. Their H¨older conjugates are denoted byq2=p1/(p11) and q1=p2/(p21) respectively. Fori = 1,2,Wper1,pi(Y) denotes the set of all functions u∈W1,pi(Y) with mean value zero that have the same trace on the opposite faces of Y. Each functionu∈Wper1,pi(Y) can be extended by periodicity to a function ofWloc1,pi(Rn).

The Euclidean norm and the scalar product in Rn are denoted by |·|and (·,·), respectively. IfA⊂Rn,|A| denotes the Lebesgue measure andχA(x) denotes its characteristic function.

The constitutive law for the -periodic composite is described by A(x, ξ) =A(x/, ξ), for every >0, for everyx∈Ω, and for everyξ∈Rn.

We have [7] thatAfulfills the following conditions:

(1) For allξ∈Rn,A(·, ξ) isY-periodic and Lebesgue measurable.

(2) |A(y,0)|= 0 for ally∈Rn.

(3) Continuity: for almost everyy∈Rn and for everyξiRn (i= 1,2) we have

|A(y, ξ1)−A(y, ξ2)| ≤C|ξ1−ξ2|α(y)(1 +1|+2|)p(y)−1−α(y) (2.1) whereα(y) =χ1(y)α1(y) +χ2(y)α2(y) and

αi(y) =

pi1 if 1≤pi2 1 ifpi2.

(4) Monotonicity: for almost everyy∈Rn and for everyξiRn (i= 1,2) we have (A(y, ξ1)−A(y, ξ2), ξ1−ξ2)≥C|ξ1−ξ2|β(y)(|ξ1|+2|)p(y)−β(y)

≥C|ξ1−ξ2|β(y)(1 +1|+2|)p(y)−β(y) (2.2) whereβ(y) =χ1(y)β1(y) +χ2(y)β2(y) and

βi(y) =

2 if 1≤pi2 pi ifpi2.

The structure conditions forA given by (2.1) and (2.2) recover the ones stated in [15] whereαi(y) = 1 and βi(y) =pifori= 1,2. In the context of this paper, the analysis when the exponentsp1andp2are in the regime between 1 and 2 becomes more involved. This particular set of structure conditions (or related variants) are used, for example, in [5,6,9,10].

2.2. Dirichlet boundary value problem

We consider the following Dirichlet boundary value problem div (A(x,∇u)) =f onΩ,

u∈W01,p1(Ω). (2.3)

wheref ∈W−1,q2(Ω).

The following homogenization result holds.

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Theorem 2.1 (homogenization theorem (see Chap. 15 of [30])). As 0, the solutions u of (2.3)converge weakly touinW1,p1(Ω), whereuis the solution of

div (b(∇u)) =f onΩ, (2.4)

u∈W01,p1(Ω); (2.5)

and the function b:Rn Rn is defined for all ξ∈Rn by b(ξ) =

YA(y, P(y, ξ))dy, (2.6)

wherep:Rn×Rn Rn is defined by

P(y, ξ) =ξ+∇υξ(y), (2.7)

whereυξ is the solution to the cell problem:

⎧⎨

Y

(A(y, ξ+∇υξ),∇w) dy= 0, for everyw∈Wper1,p1(Y),

υξ∈Wper1,p1(Y). (2.8)

Remark 2.2. The followinga priori bound is satisfied sup>0 Ω

χ1(x)|∇u(x)|p1dx+

Ω

χ2(x)|∇u(x)|p2dx

≤C <∞, (2.9)

whereC does not depend on. The proof of this bound is given in Lemma3.5.

Remark 2.3. The functionb, defined in (2.6), is continuous and monotone (see Lem.3.7).

Remark 2.4. Since the solutionυξof (2.8) can be extended by periodicity to a function ofWloc1,p1(Rn), then (2.8) is equivalent to−div(A(y, ξ+∇υξ(y))) = 0 overD(Rn),i.e.,

−div (A(y, P(y, ξ))) = 0 inD(Rn) for everyξ∈Rn. (2.10) Moreover, by (2.8), we have

Y

(A(y, P(y, ξ)), P(y, ξ)) dy=

Y

(A(y, P(y, ξ)), ξ) dy = (b(ξ), ξ). (2.11) For >0, defineP:Rn×Rn Rn by

P(x, ξ) =Px , ξ

=ξ+∇υξx

, (2.12)

whereυξ is the unique solution of (2.8). The functionsP andP are easily seen to have the following properties P, ξ) isY-periodic andP(x, ξ) is-periodic inx, (2.13)

Y

P(y, ξ)dy=ξ, (2.14)

P, ξ) ξ inLp1(Ω;Rn) as0, (2.15)

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P(y,0) = 0 for almost every y, (2.16)

, P, ξ)

b(ξ) inLq2(Ω;Rn), as 0. (2.17) We now state the higher order integrability properties of the homogenized solution for periodic dispersions of inclusions and layered microgeometries.

Theorem 2.5. Given a periodic dispersion of inclusions or a layered material then the solution u of (2.4) belongs toW01,p2(Ω).

Remark 2.6. The proof of Theorem 2.5 [15] uses a variational approach and considers the homogenized Lagrangian associated with b(ξ) defined in (2.6). The integrability of the homogenized solution u of (2.4) is determined by the growth of the homogenized Lagrangian with respect to its argument, which follows from the regularity of the Lagrangian. For periodic dispersed and layered microstructures no Lavrentiev phenomenon occurs. The proof of the regularity of the homogenized Lagrangian for periodic dispersed microstructure can be found in Chapter 14 of [30] and for layered microstructure in [15]. Both proofs are valid forp1, p21

2.2.1. Statement of the corrector theorem

We now describe the family of correctors that provide a strong approximation of the sequencei∇u}>0 in theLpi(Ω,Rn) norm, fori= 1,2. We denote the rescaled period cell with side length >0 byY and write Yk =k+Y, wherek∈Zn. In what follows it is convenient to define the index setI=

k∈Zn:Yk⊂Ω . Forϕ∈Lp2(Ω;Rn), we define the local average operatorM associated with the partitionYk, k∈Iby

M(ϕ)(x) =

k∈I

χYk (x) 1

|Yk|

Yk

ϕ(y)dy; if x∈

k∈I

Yk. (2.18)

The family of approximations of the identity map M has the following properties (for a proof, see, for example [29]):

1. Fori= 1,2, M(ϕ)−ϕ Lpi(Ω;Rn)0 as0.

2. M(ϕ)→ϕa.e. onΩ.

3. From Jensen’s inequality we have M(ϕ) Lpi(Ω;Rn)≤ ϕ Lpi(Ω;Rn), for everyϕ∈Lp2(Ω;Rn) andi= 1,2.

The strong approximation to the sequencei∇u}>0 is given by the following corrector theorem.

Theorem 2.7 (corrector theorem).Let f ∈W−1,q2(Ω), let u be the solutions to the problem (2.3), and letu be the solution to problem(2.4). Then, up to a subsequence, for periodic dispersions of inclusions and for layered materials, we have

Ωi(x)P(x, M(∇u)(x))−χi(x)∇u(x)|pidx0, (2.19) as→0, for i= 1,2.

The proof of Theorem2.7is given in Section4.1.

2.2.2. Lower bounds on the local amplification of the macroscopic field

We show lower bounds on the Lq norm of the gradient fields inside each material that are given in terms of the correctors presented in Theorem 2.7. We begin by presenting a general lower bound that holds for the composition of the sequencei∇u}>0with any non-negative Carath´eodory function. Recall thatψ:Ω×Rn Ris a Carath´eodory function if ψ(x,·) is continuous for almost everyx∈Ω and if ψ(·, λ) is measurable in x for everyλ∈Rn. The lower bound on the sequence obtained by the composition ofψ(x,·) withχi(x)∇u(x) is given by

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Theorem 2.8. For all Carath´eodory functionsψ≥0 and measurable setsD⊂Ωwe have

D

Y

ψ(x, χi(y)P(y,∇u(x))) dydx≤lim inf

→0

D

ψ(x, χi(x)∇u(x)) dx.

If the sequence{ψ(x, χi(x)∇u(x))}>0 is weakly convergent inL1(Ω), then the inequality becomes an equal- ity.

In particular, forψ(x, λ) =|λ|q with q >1, we have

D

Y

χi(y)|P(y,∇u(x))|qdydxlim inf

→0

D

χi(x)|∇u(x)|qdx. (2.20)

Theorem 2.8 together with (2.20) provide explicit lower bounds on the gradient field inside each material.

It relates the local excursions of the gradient inside each phase χi∇u to the average gradient ∇u through the multiscale quantity given by the corrector P(y,∇u(x)). It is clear from (2.20) that the Lq(Y ×Ω;Rn) integrability ofP(y,∇u(x)) provides a lower bound on theLq(Ω;Rn) integrability of∇u.

The proof of Theorem2.8is given in Section4.2.

3. Technical lemmas

In this section we state some technical a priori bounds and convergence properties for the sequencesPdefined in (2.12),∇u, andA(x, P(x,∇u)) that are used in the proof of the main results of this paper. The proofs of these lemmas can be found in the Appendix.

Lemma 3.1. For every ξ∈Rn we have

Y

χ1(y)|P(y, ξ)|p1dy+

Y

χ2(y)|P(y, ξ)|p2dy≤C(1 +|ξ|p1θ1+|ξ|p2θ2) (3.1) and by a change of variables, we obtain

Y

χ1(x)|P(x, ξ)|p1dx+

Y

χ2(x)|P(x, ξ)|p2dx≤C(1 +|ξ|p1θ1+|ξ|p2θ2)|Y|. (3.2)

Lemma 3.2. For every ξ1, ξ2Rn we have – For1< p1≤p22:

Y

χ1(x)|P(x, ξ1)−P(x, ξ2)|p1dx+

Y

χ2(x)|P(x, ξ1)−P(x, ξ2)|p2dx (3.3)

≤C

θ13−p11 1−ξ2|3−pp11 (1 +1|p1θ1+2|p1θ1+1|p2θ2+2|p2θ2)

2−p1 3−p1

+θ22p2−pp11p2+p1 1−ξ2|2p2−pp11pp22+p1 (1 +1|p1θ1+2|p1θ1+1|p2θ2+2|p2θ2)2pp2−p2(21−pp2+1)p1 +θ12p1−pp12p2+p2 1−ξ2|2p1−pp11pp22+p2 (1 +1|p1θ1+2|p1θ1+1|p2θ2+2|p2θ2)

p1(2−p2) 2p1−p1p2+p2

+θ23−p12 1−ξ2|3−pp22(1 +1|p1θ1+2|p1θ1+1|p2θ2+2|p2θ2)

2−p2 3−p2

|Y|;

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– For1< p12≤p2:

Y

χ1(x)|P(x, ξ1)−P(x, ξ2)|p1dx+

Y

χ2(x)|P(x, ξ1)−P(x, ξ2)|p2dx (3.4)

≤C

θ13−p11 1−ξ2|3−pp11 (1 +1|p1θ1+2|p1θ1+1|p2θ2+2|p2θ2)

2−p1 3−p1

+ θ22p2p−p1 1 1−ξ2|2pp21−pp21 (1 +1|p1θ1+2|p1θ1+1|p2θ2+2|p2θ2)2(p2p22−p−p1)1 +θ11−ξ2|p1 +θ2p21−1 1−ξ2|p2p−12 (1 +1|p1θ1+2|p1θ1+1|p2θ2+2|p2θ2)pp22−2−1

|Y|.

Remark 3.3. Note the two “extra” terms in (3.3) of Lemma3.2where there is a “mixing” of the exponentsp1 andp2which do not appear in the corresponding property ofP given by Lemma 5.2 in [15]. Since Lemma3.2 is used to prove Lemma 3.4, these two terms appear again in (3.6) and therefore in the proof of Theorem2.7 and the proof of Lemma4.1 used to prove Theorem2.8.

Lemma 3.4. Let ϕbe such that

sup>0

Ω

χ1(x)|ϕ(x)|p1dx+

Ω

χ2(x)|ϕ(x)|p2dx

<∞,

and letΨ be a simple function of the form

Ψ(x) =m

j=0

ηjχΩj(x), (3.5)

with ηj Rn\ {0}, Ωj ⊂⊂ Ω, |∂Ωj| = 0, Ωj ∩Ωk = for j = k and j, k = 1, . . . , m; and set η0 = 0 and Ω0=Ω\ m

j=1

Ωj. Then

– For1< p1≤p22:

lim sup

→0

2 i=1

Ω

χi(x)|P(x, Mϕ)−P(x, Ψ)|pidx

≤Clim sup

→0 Ω

χ1(x)|ϕ−Ψ|p1dx 3−p1

1 |Ω|+

Ω

χ1(x)|ϕ|p1dx

+

Ωχ2(x)|ϕ|p2dx+

Ωχ1(x)|Ψ|p1dx+

Ωχ2(x)|Ψ|p2dx 2−p3−p1

1

+

Ωχ2(x)|ϕ−Ψ|p2dx p p1

2−p1p2+p1

|Ω|+

Ωχ1(x)|ϕ|p1dx+

Ωχ2(x)|ϕ|p2dx +

Ω

χ1(x)|Ψ|p1dx+

Ω

χ2(x)|Ψ|p2dx

pp2(2−p1)

2−p1p2+p1

(3.6)

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+

Ω

χ1(x)|ϕ−Ψ|p1dx p p2

1−p1p2+p2

|Ω|+

Ω

χ1(x)|ϕ|p1dx+

Ω

χ2(x)|ϕ|p2dx

+

Ω

χ1(x)|Ψ|p1dx+

Ω

χ2(x)|Ψ|p2dx

pp1(2−p2)

1−p1p2+p2

+

Ω

χ2(x)|ϕ−Ψ|p2dx 3−p1

2 |Ω|+

Ω

χ1(x)|ϕ|p1dx+

Ω

χ2(x)|ϕ|p2dx

+

Ω

χ1(x)|Ψ|p1dx+

Ω

χ2(x)|Ψ|p2dx 2−p3−p2

2

; – For1< p12≤p2:

lim sup

→0

2 i=1

Ω

χi(x)|P(x, Mϕ)−P(x, Ψ)|pidx

≤Clim sup

→0 Ω

χ1(x)|ϕ−Ψ|p1dx 3−p1

1 |Ω|+

Ω

χ1(x)|ϕ|p1dx

+

Ω

χ2(x)|ϕ|p2dx+

Ω

χ1(x)|Ψ|p1dx+

Ω

χ2(x)|Ψ|p2dx 2−p3−p1

1 (3.7)

+

Ω

χ2(x)|ϕ−Ψ|p2dx 2pp1

2−p1

|Ω|+

Ω

χ1(x)|ϕ|p1dx+

Ω

χ2(x)|ϕ|p2dx

+

Ω

χ1(x)|Ψ|p1dx+

Ω

χ2(x)|Ψ|p2dx

2(p2p2−p1)

2−p1

+

Ω

χ1(x)|ϕ−Ψ|p1dx+

Ω

χ2(x)|ϕ−Ψ|p2dx p1

2−1

|Ω|+

Ω

χ1(x)|ϕ|p1dx

+

Ω

χ2(x)|ϕ|p2dx+

Ω

χ1(x)|Ψ|p1dx+

Ω

χ2(x)|Ψ|p2dx pp2−2

2−1 .

Lemma 3.5. Let u be the solution to(2.3). Then the a prioribound(2.9) holds.

Lemma 3.6. If the microstructure is dispersed or layered, we have that sup>0

Ω

χi(x)|P(x, M∇u(x))|pidx

≤C <∞, for i= 1,2.

We use Lemma3.1to prove structure conditions ofb (2.6) in the following lemma.

Lemma 3.7. The functionb, defined in(2.6), satisfies the following structure properties: for everyξ1, ξ2Rn 1. Monotonicity:

(b(ξ2)−b(ξ1), ξ2−ξ1)0. (3.8)

2. Continuity: there exists a positive constantC such that – For1< p1≤p22:

|b(ξ1)−b(ξ2)| ≤C

2−ξ1|p3−p1−11 (1 +θ11|p1+θ21|p2+θ12|p1+θ22|p2)

(2−p1)(p1−1) p1(3−p1)

+2−ξ1|p3−p2−12 (1 +θ11|p1+θ21|p2+θ12|p1+θ22|p2)

(2−p2)(p2−1) p2(3−p2)

; (3.9)

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– For1< p12≤p2:

|b(ξ1)−b(ξ2)| ≤C

2−ξ1|p3−p1−11 (1 +θ11|p1+θ21|p2+θ12|p1+θ22|p2)

(2−p1)(p1−1) p1(3−p1)

+2−ξ1|p21−1(1 +θ11|p1+θ21|p2+θ12|p1+θ22|p2)pp22−2−1

. (3.10)

These structure conditions for b are different to the ones obtained in [15] and their proofs require different techniques, for example, the use of(A.4)to obtain(3.9). These structure conditions(3.9)and(3.10)will be used in the proof of Theorem 2.7.

Lemma 3.8. For all j= 0, . . . , m, we have that

Ωj

|(A(x, P(x, ηj)),∇u(x))|dxand

Ωj

A(x,∇u(x)), P(x, ηj)dxare uniformly bounded with respect to .

Lemma 3.9. As 0, up to a subsequence, (A, P, ηj)),∇u(·)) converges weakly to a function gj L1j;R), for all j = 0, . . . , m. In a similar way, up to a subsequence, (A,∇u(·)), P, ηj)) converges weakly to a function hj ∈L1j;R), for allj= 0, . . . , m.

4. Proof of main results

4.1. Proof of the corrector theorem

We are now in the position to give the proof of Theorem 2.7. We present the proof for the case when 1< p1 ≤p22, for 1< p12 ≤p2 the proof is very similar and the correspondig formulas can be found in the appendix, in SectionA.9

Proof. Letu∈W01,p1(Ω) be the solutions of (2.3). By (2.2), Lemmas 3.5and3.6we have that

Ω

1(x)|P(x, M∇u(x))− ∇u(x)|p1+χ2(x)|P(x, M∇u(x))− ∇u(x)|p2] dx

≤C |Ω|+

Ω

χ1(x)|P(x, M∇u(x))|p1dx+

Ω

χ1(x)|∇u(x)|p1dx 2−p21

× Ωχ1(x) (A(x, P(x, M∇u(x)))−A(x,∇u(x)), P(x, M∇u(x))− ∇u(x)) dx p21

+

Ωχ2(x) (A(x, P(x, M∇u(x)))−A(x,∇u(x)), P(x, M∇u(x))− ∇u(x)) dx p22

× Ω

χ2(x) (1 +|P(x, M∇u(x))|p2+|∇u(x)|)p2dx 2−p22

≤C2

i=1 Ω

(A(x, P(x, M∇u(x)))−A(x,∇u(x)), P(x, M∇u(x))− ∇u(x)) dx pi2

.

(11)

To prove Theorem2.7, we show that

Ω

(A(x, P(x, M∇u(x)))−A(x,∇u(x)), P(x, M∇u(x))− ∇u(x)) dx

=

Ω

(A(x, P(x, M∇u)), P(x, M∇u)) dx−

Ω

(A(x, P(x, M∇u)),∇u) dx

Ω

(A(x,∇u), P(x, M∇u)) dx+

Ω

(A(x,∇u),∇u) dx goes to 0, as0. This is done in four steps.

In what follows, we use the following notation ξk= 1

|Yk|

Yk∇udx.

Step 1.Let us prove that

Ω

(A(x, P(x, M∇u)), P(x, M∇u)) dx→

Ω(b(∇u),∇u) dx (4.1) as0.

Proof. From (2.11) and (2.18), we obtain

Ω

(A(x, P(x, M∇u(x))), P(x, M∇u(x))) dx=

k∈I

Yk

Ax , Px

, ξk , Px

, ξk dx

=n

k∈I

Y

A y, P

y, ξk , P

y, ξk dy

=

k∈I

Ω

χYk (x)

b(ξk), ξk dx

=

Ω

(b(M∇u(x)), M∇u(x)) dx.

By (3.9) in Lemma3.7, H¨older’s inequality, Theorem2.5, and Jensen’s inequality, we have

Ω|b(M∇u(x))−b(∇u(x))|q1dx≤C

Ω|M∇u− ∇u|p2dx

(p p1−1

2−1)(3−p1 )

+

Ω|M∇u− ∇u|p2dx 3−p1

2

.

From Property 1 ofM, we obtain that

b(M∇u)→b(∇u) inLq1(Ω;Rn), as 0. (4.2) Now, (4.1) follows from (4.2) since M∇u→ ∇uin Lp2(Ω;Rn), so

Ω(A(x, P(x, M∇u(x))), P(x, M∇u(x))) dx=

Ω(b(M∇u(x), M∇u(x)) dx

Ω(b(∇u(x)),∇u(x)) dx, as→0.

Références

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