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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Characterizing the strange term in critical size homogenization: Quasilinear equations with a general microscopic boundary condition

Jesus Ildefonso DíazORCID iD: https://orcid.org/0000-0003-1730-9509 / David Gómez-Castro
  • Departamento de Matemática Aplicada e I.M.I., Universidad Complutense de Madrid, Plaza de las Ciencias 3, 28040 Madrid, Spain
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  • De Gruyter OnlineGoogle Scholar
/ Alexander V. Podol’skii / Tatiana A. Shaposhnikova
Published Online: 2017-08-03 | DOI: https://doi.org/10.1515/anona-2017-0140

Abstract

The aim of this paper is to consider the asymptotic behavior of boundary value problems in n-dimensional domains with periodically placed particles, with a general microscopic boundary condition on the particles and a p-Laplace diffusion operator on the interior, in the case in which the particles are of critical size. We consider the cases in which 1<p<n, n3. In fact, in contrast to previous results in the literature, we formulate the microscopic boundary condition in terms of a Robin type condition, involving a general maximal monotone graph, which also includes the case of microscopic Dirichlet boundary conditions. In this way we unify the treatment of apparently different formulations, which before were considered separately. We characterize the so called “strange term” in the homogenized problem for the case in which the particles are balls of critical size. Moreover, by studying an application in Chemical Engineering, we show that the critically sized particles lead to a more effective homogeneous reaction than noncritically sized particles.

Keywords: Homogenization; nonlinear boundary reaction; noncritical sizes, maximal monotone graphs

MSC 2010: 35B25; 35B40; 35J05; 35J20

1 Introduction

A well-known effect in homogenization theory is the appearance of some changes in the structural modelling of the homogenized problem for suitable critical size of the elements configuring the “micro-structured” material. It seems that the first result in this direction was presented in the pioneering paper by Marchenko and Hruslov [27]. A more popular presentation of the appearance of some “strange terms” was due to Cioranescu and Murat [4]. Both articles dealt with linear equations with Neumann and Dirichlet boundary conditions, respectively. Since then many papers were devoted to different formulations, e.g., more general elliptic partial differential equations (possibly of quasilinear type), Robin type and other boundary conditions of different nature, etc. It is impossible to mention all of them here (a few of them will be mentioned in the rest of the introduction) but the reader may imagine that the nature of this “strange term” may be completely different according to the peculiarities of the formulation in consideration (something that was already indicated at the end of the introduction of the paper by Cioranescu and Murat [4]).

The main goal of this paper is to characterize the change of structural behavior arising in the homogenization process when applied to chemical reactions taking place on fixed-bed nanoreactors, at the microscopic level, on the boundary of the particles

{-Δpuε=f(x),xΩε,-νpuεε-γσ(uε),xSε,uε=0,xΩ(1.1)

for a very general type of chemical kinetics (here given by the maximal monotone graph σ of 2). Thanks to this generality on the maximal monotone graph σ, our treatment also includes the case of microscopic Dirichlet boundary conditions. In this way we unify the treatment of apparently different formulations, which before were considered separately.

The diffusion is modeled by the quasilinear operator Δpuεdiv(|uε|p-2uε), with p>1. Notice that p=2 corresponds to the linear diffusion operator, and that p2 appears in turbulent regime flows or non-Newtonian flows (see [8]). As it is well known, this operator appears in many other contexts and is one of the best examples of quasilinear operators leading to a formulation in terms of nonlinear monotone operators (see, e.g., [26, 1, 7]).

The “normal derivative” must be then understood as νpuε=|uε|p-2uε𝝂, where 𝝂 is the outward unit normal vector on the boundary of the particles SεΩε. In fact, we will consider the structural assumption

1<p<n,n3.

The cases pn are completely different, see [31, 32] (see also, for instance, the study made for a general monotone quasilinear equation with Dirichlet boundary conditions in [7]).

As mentioned before, the generality assumed on the maximal monotone graph σ of 2 allows to treat, in a unified way, different cases as the case of Dirichlet boundary conditions, which corresponds to the choice of σ given by

D(σ)={0}andσ(0)=(-,+)(1.2)

(see, e.g. [1]), and the case of nonlinear Robin type boundary conditions, which corresponds (see, e.g. [24]) to the case in which D(σ)= and σ is a continuous nondecreasing function.

The domain Ωεn is assumed to have an ε-periodical structure. Since our main goal is to get a very precise description of the so-called “strange term” in the homogenized problem, we shall assume that the particles are balls of radius aε=C0εα, where α>1. One of the interesting properties that arise from our precise characterization is that there is uniqueness of solutions of the homogenized problem. This was not always proved in previous results (cf. the general framework considered in [7], and how their characterization, given in their Lemma 5.1, is not enough to get the uniqueness of solution of their homogenized problem). The consideration of particles of a general shape is a difficult task, especially the exact identification of the “strange terms”. A similar formulation to the one considered in this paper for that case can be obtained, at least for continuous σ, and has been the subject of a different paper (see [15]).

The problem has two different parameters: α, the size of the particles, and γ, the normalization factor of the boundary condition on Sε. When they have critical values

α=nn-p,γ=α(n-1)-n=α(p-1),

then our main result in this paper shows that the homogenized problem involves a different distributed chemical kinetics nonlinearity:

{-Δpu+𝒜|H(u)|p-2H(u)=f(x)in Ω,u=0on Ω,(1.3)

where

𝒜=(n-pp-1)p-1C0n-pωn(1.4)

and H: is given by

H(r)=(I+σ-1Θn,p)-1(r),(1.5)

with

Θn,p(s)=0|s|p-2sfor s

and

0=(n-pC0(p-1))p-1,(1.6)

where ωn is the surface area of the unit sphere in n. We show that, for any maximal monotone graph σ, H is a nondecreasing contraction, and thus the existence, uniqueness and continuous dependence of solutions of the homogenized problem is consequence of well-known results on monotone operator theory.

The change of behavior from the nonlinearity of type σ in the nonhomogeneous problem to the nonlinearity H in the homogeneous problem is one of the characteristics of the nanotechnological effects (see, e.g., [33]) and does not appear if 1α<nn-p (see [5, 34]).

Before presenting the details of the notation used above, let us mention that our main aim is to provide a common roof and extend (under different points of view) some previous results in the literature concerning different structural assumptions (i.e., the functions σ and H) after the homogenization process.

The case of Robin boundary conditions nu+β(ε)σ(u)=0 on Sε was first studied by Marchenko and Hruslov in a series of papers dealing mainly with the linear case σ(u)=λu, see [19, 20, 21, 27]. Some references on different choices of smooth functions σ can be found in [6, 18, 24, 25, 37, 22, 29] and the references therein. For further references, see [13, 23, 17, 16]. Some previous results by the authors [11], formulated there for some not necessarily Lipschitz functions σ and p[2,n), will be here extended to the case a general maximal monotone graph σ (which includes the case of Dirichlet boundary conditions) and p(1,n).

The special case of Dirichlet boundary condition uε=0 on Sε, covered by (1.2), gives σ-1(s)=0 for any s, and so H(r)=r for any r. Therefore, the “strange term” arising in the homogenized equation becomes 𝒜|u|p-2u. This was shown for p=2 in the pioneering paper by Cioranescu and Murat [4]. However, even in this simple case, the treatment in [7] for the case p2 is not as sharp as in our case. In [7], Dal Maso and Skrypnik do not provide an explicit expression for this strange term. In fact, their characterization (see [7, Lemma 5.1]) does not guaranty uniqueness of solutions of the homogenized problem.

The case of the boundary condition

uε0,nuε+ε-γσ0(uε)0,uε(nuε+ε-γσ0(uε))=0on Sε,

which was studied for smooth σ0 in [22] by ad hoc techniques, is also covered by the common proof provided in this paper, by taking

D(σ)=[0,+),σ(u)={(-,0]if u=0,σ0(u)if u>0.

See also [12, 28].

The choice of the critical values of α and γ might appear arbitrary. Let us give some reasons why this is a good choice. First, if N(ε) is the number of particles, then N(ε)ε-n. It is easy to see that |Sε|=N(ε)|(aεG0)|εα(n-1)-n, where G0 is the unit ball centered at 0. Let us analyze the choice of γ. If we consider the reaction term on the weak formulation, with σ(uε) a bounded sequence in L and v a bounded test function, then

1|Sε|Sεσ(uε)v𝑑Sε-(α(n-1)-n)Sεσ(uε)v𝑑S

is a bounded sequence. Hence, if the sequence uε is bounded in L and v is a bounded test function, then

ε-γSεσ(uε)v𝑑S

can only be expect to tend to either 0 or + if γα(n-1)-n, and hence we will lose the reaction term on the equation on the homogenized equation or we lose the equation altogether. If the macroscopic behavior is given by a reaction diffusion equation (with nontrivial reaction), then the choice scaling γ as ε0 can be no other.

The appearance of the critical value of α has to do with a property of traces. It is known (see [30]) that

aεS0|u|pdSK(aεn-1ε-nYε|u|pdx+aεp-1Yε|u|pdx).

As it turns out, the critical scale is the one in which both terms in the right-hand side have the same order of convergence. Notice that, in the critical case α=nn-p, we have γ=α(p-1).

Notice that for a Newtonian fluid in 3 (n=3, p=2), the critical size corresponds to α=3. Obviously the critical value of α is an increasing function of p. Therefore, for non-Newtonian dilatant fluids or a Newtonian flow in turbulent regime (p>2), our assumption means α>3, the particles are tiny with respect to their repetition, whereas for pseudoplastic fluids (p<2), the critical particles satisfy α<3, and hence are not so tiny with respect to their repetition.

A relevant application of our results is the following. Let us consider the usual formulation in Chemical Engineering (see [35, 9]) with a constant external supply

{-Δwε=0,xΩε,νwε+ε-γg(wε)=0,xSε,wε=1,xΩ,

where g is a nondecreasing real function such that g(0)=0. In order to adapt our results, we introduce the change in variable u=1-w and σ(u)=g(1)-g(1-u), and the problem becomes

{-Δuε=0,xΩε,νuε+ε-γσ(uε)=ε-γg(1),xSε,uε=0,xΩ.

Notice that the presence of wε=1 on Ω is translated to a source in Sε for uε. We will see later (Theorem 6.2) that the new equation for H, when α=nn-2, is

n-2C0H(s)=σ(s-H(s))-g(1),(1.7)

that is,

H(u)=-(g-1(n-2C0)+Id)-1(1-u),

so that an extension of wε converges weakly in H1(Ω) to wcrit, the solution of

{-Δwcrit+𝒜h(wcrit)=0in Ω,wcrit=1on Ω,

and h is given by

h(w)=(g-1(n-2n)+Id)-1(w).

Notice that in the case of Neumann problems, σ(s)0 for any sR, and although σ-1 is a well-known maximal monotone graph, the more direct identification of the “strange term” H(u) is obtained trough the implicit equation (1.7), since in this case we get that

H(s)=-C0g(1)n-2for any s.

In the noncritical cases, 1<α<nn-2, we will show that an extension of wε converges weakly in H1(Ω) to wnon-crit, the solution of

{-Δwnon-crit+𝒜^g(wnon-crit)=0in Ω,wnon-crit=1on Ω,

with 𝒜^=C0n-1|G0|. Finally, we will show in Theorem 6.3 that

wcritwnon-crit,

so we have a pointwise “better” reaction in the critical case [10]. We point out that a different criterion to establish the optimality of the reaction in terms of the so-called “chemical effectiveness” was considered by the authors in [14].

The plan of the rest of the paper is the following: Section 2 will be devoted to the statement of the main results, Section 3 contains the proof of the existence results for equation (1.1) and the characterization of H, Section 4 is devoted to the proof of the main result, Theorem 2.4, and Section 5 contains the proof of the auxiliary Theorem 2.9, which studies the limit of the diffusion. We conclude the paper with Section 6, where we study the noncritical case and the pointwise comparison of its homogenized solution with the critical case.

2 Statement of the main results

Let Ω be a bounded domain in n, n3, with a smooth boundary Ω, and let Y=(-12,12)n. Denote by G0=B1(0) the unit ball centered at the origin. This plays a crucial role in the proof. As far as we known, no results are known in the critical cases if G0 is not a ball. For δ>0 and ε>0, we define sets δB={x:δ-1xB} and Ω~ε={xΩ:ρ(x,Ω)>2ε}. Let

aε=C0εα,

where α>1 and C0 is a given positive number. Define

Gε=jΥε(aεG0+εj)=jΥεGεj,

where Υε={jn:(aεG0+εj)Ω~¯ε}, N(ε)=|Υε|ε-n, and n denotes the set of vectors z with integer coordinates. Define Yεj=εY+εj, where jΥε and note that G¯εjY¯εj and center of the ball Gεj coincides with the center of the cube Yεj. Our “microscopic domain” is defined as

Ωε=ΩGε¯,Sε=Gε,Ωε=ΩSε.

We define the space W01,p(Ωε,Ω) as the completion, with respect to the norm of W1,p(Ωε), of the set of infinitely differentiable functions in Ω¯ε equal to zero in a neighborhood of Ω, that is,

W01,p(Ωε,Ω)={uW1,p(Ωε):u=0 on Ω}.

Concerning the solvability of problem (1.1), we start by introducing the notion of weak solution. Since we assume that σ:𝒫(), where 𝒫() denotes the set of subsets of , we recall, by well-known results (see, e.g., [2]), that

σ is a maximal monotone graph of 20σ(0),(2.1)

and that there exists a convex lower semicontinuous function Ψ:(-,+], with Ψ(0)=0, such that σ=Ψ is its subdifferential. We also know that if we define

D(σ)={r such that σ(r)},

where denotes the empty set, and

D(Ψ)={r such that Ψ(r)<+},

then D(σ)D(Ψ)D(Ψ)¯=D(σ)¯.

In the rest of the paper we will always assume that fLp(Ω), where, as usual, p=pp-1.

Since uε is the minimizer of the following energy functional in W1,p(Ωε,Ω) (see [26, 1]):

E(u)=Ωε|u|pdx+ε-γSεΨ(u)dS-Ωεfudx,

we consider the following definition of weak solution.

Definition 2.1.

We will say that uεW1,p(Ωε,Ω) is a weak solution of problem (1.1) if uε(x)D(Ψ) for a.e. xSε, and for all vW1,p(Ωε,Ω), we have

Ωε|uε|p-2uε(v-uε)dx+ε-γSε(Ψ(v)-Ψ(uε))dSΩεf(v-uε)dx.(2.2)

The existence and uniqueness of a weak solution to problem (2.2) is an easy consequence of well-known results:

Proposition 2.2.

There exists a unique uεW1,p(Ωε,Ω) weak solution of (2.2). Besides, there exists K>0 independent of ε such that

uεLp(Ωε)+ε-γΨ(uε)L1(Sε)K.(2.3)

The homogenized problem will involve the function H: given by (1.5). Let us present some of the properties satisfied by H.

Lemma 2.3.

If σ satisfies (2.1), then the function H defined by (1.5) is a nondecreasing nonexpansion on R (i.e., a nondecreasing Lipschitz continuous function of Lipschitz constant L1). Moreover, this function H is the unique function H:RR satisfying the relation

0|H(r)|p-2H(r)σ(r-H(r))for any r.(2.4)

Concerning the homogenized problem (1.3), we point out that since H is a nondecreasing nonexpansion on , for the parameters 𝒜 and 0 given by (1.4) and (1.6), and for fLp(Ω), there exists a unique weak solution uW01,p(Ω) of problem (1.3). Moreover, |H(u)|p-2H(u)Lp(Ω). For the proof it is enough to set V=W01,p(Ω) and define the operator A:VV by

Av,w=Ω|v|p-2vwdx+Ω𝒜|H(v)|p-2H(v)wdxfor any wV.(2.5)

Notice that, since H is Lipschitz, H(v)Lp(Ω) for any vLp(Ω). Then A is a hemicontinuous strictly monotone coercive operator, and the existence and uniqueness of a weak solution u is standard (see, e.g., [26]).

We will make fundamental use of the following reformulation of a weak solution. Since the limit operator A:VV, with V=W01,p(Ω), given by (2.5), is hemicontinuous and monotone, we can use the Brezis–Sibony characterization (see [3, Lemma 1.1] or [26, Chapter 2, Theorem 2.2]), that is, uW01,p(Ω) is a weak solution of (1.3) if and only if

Ω|v|p-2v(v-u)dx+Ω0|H(v)|p-2H(v)(v-u)dxΩf(v-u)dxfor any vW01,p(Ω).(2.6)

The main result of this paper is the following convergence result.

Theorem 2.4.

Let n3, 1<p<n, α=nn-p and γ=α(p-1). Let σ be any maximal monotone graph of R2, with 0σ(0), and let fLp(Ω). Let uεW01,p(Ωε,Ω) be the (unique) weak solution of problem (1.1). Then there exists an extension u~ε of uε such that u~εu in W01,p(Ω) as ε0, where uW01,p(Ω) is the (unique) weak solution of problem (1.3) associated to the function H, defined by (1.5).

Remark 2.5.

The case n=2 can be studied by similar techniques, although some of the computations vary. In particular, the critical value of α does not verify the same formula.

The other key result we will prove in this paper is Theorem 2.9 below, the statement of which requires some preliminary lemmas. The extension u~ε of solutions uε can be obtained by applying the methods of [30].

Lemma 2.6.

Let Ωε be the domain defined above and let 1<p<n, n3. Then there exists an extension operator Pε:W1,p(Ωε)W1,p(Ω) such that

PεuW1,p(Ω)C1uW1,p(Ωε),(Pεu)Lp(Ω)C2uLp(Ωε).

Moreover, by applying this extension theorem and the methods introduced in [30], we can prove the following useful estimates.

Lemma 2.7.

  • (i)

    Let uW01,p(Ωε,Ω), p>1 and n3 . Then there exists positive constant C such that

    uLp(Ωε)CuLp(Ωε).

  • (ii)

    Let uW1,p(Yε) be such that Yεu=0 . Then

    uLp(Yε)K1εuLp(Yε),

    where the constant K1 is independent of ε.

Thanks to the a priori estimate (2.3) and the properties of the extension operator Pε:W01,p(Ωε,Ω)W1,p(Ω), we know that and there exists uW01,p(Ω) such that

Pεuεuin W01,p(Ω).

The difficult task is to show that uW01,p(Ω) is the weak solution of problem (1.3) such as it is ensured in Theorem 2.4.

Motivated by this and (2.6), we will also use the fact that if uεW01,p(Ωε,Ω) is the weak solution of problem (1.1), then

Ωε|v|p-2v(v-uε)dx+ε-γSε(Ψ(v)-Ψ(uε))dSΩεf(v-uε)dx(2.7)

for any test function vW1,p(Ωε,Ω).

The problematic term, in order to pass to the limit, is the boundary integrals over Sε. Here we will follow a technique of proof introduced by the last author (Shaposhnikova) in collaboration with different co-authors (see, e.g., [28, 37, 34]), which can be applied in different frameworks.

Lemma 2.8.

Let zεW01,p(Ω) for some p>1, and assume that zεz0 in W01,p(Ω) as ε0. Then

|22(n-1)εjΥεTε/4jzε𝑑S-ωnΩz0𝑑x|0as ε0,

where ωn is the surface area of the unit sphere in Rn.

This lemma (which we remark is independent of α and γ, see the proof in [37]) is the key point of the homogenization technique in the critical case. It is based in the general idea that if Pεj is the center of the ball Gεj={xYεj:|x-Pεj|<aε} and if Tεj denotes the ball of radius ε/4 centered at the point Pεj, then we can get several explicit estimates on the solution wεj(x) for j=1,,N(ε) of the auxiliary cellular boundary value problem

{Δpwεj=0,xTεjGεj¯,wεj=1,xGεj,wεj=0,xTεj.(2.8)

One of the many remarkable properties of this cellular problem is that its (unique) weak solution, wεj, is radially symmetric (recall that G0 is a ball) and satisfies that νpwεj is constant on Tεj and on Gεj. Due to the divergence theorem,

Gεj|wεj|p-2wεjzdx=TεjzνpwεjdS+GεjzνpwεjdSfor any zW1,p(TεjGεj¯).

Furthermore, we can make explicitly several computations. Hence, we have an explicit way to compare the reaction term on Sε with an auxiliary term on balls with radius Cε, and Lemma 2.8 becomes very useful.

Another key idea of our proof is to relate a general test function vW01,p(Ω), used to check the limit characterization (2.6), with some suitable correction vε, which is a better fitted test function in the microscopic weak formulation (2.7). In fact, by density, it will be enough to do that with a smooth test function v𝒞c(Ω). We will construct such adaptation among test functions in the form vε=v-hWε, where, for the moment, hW1,(Ω) without any other property, and, which is crucial, WεW01,(Ω) defined as

Wε={wεj,xTεjGεj¯,j=1,,N(ε)=|Υε|,1,xGε,0,xnj=1N(ε)Tεj,(2.9)

with wεj the solution of the auxiliary cellular boundary value problem (2.8). The following technical result will explain why the function H, arising in the limit problem (1.3), was taken in this concrete form (more precisely, so that (2.4) holds), different from the boundary kinetics σ.

Theorem 2.9.

Let uεW01,p(Ωε,Ω), 1<p<n, be a sequence of uniformly bounded norm, and let vCc(Ω), hW1,(Ω) and vε=v-hWε. Then

limε0(Ωε|vε|p-2vε(vε-uε)dx)=limε0(I1,ε+I2,ε+I3,ε),

where

I1,ε=Ωε|v|p-2v(v-uε)dx,(2.10)I2,ε=-ε-γ0Sε|h|p-2h(v-h-uε)dS,I3,ε=-AεεjΥεTεj|h|p-2h(v-uε)dS,(2.11)

with Aε being a bounded sequence, see (5.1). Besides, if u~ε is an extension of uε and u~εu in W01,p(Ω), then, for any vW01,p(Ω),

limε0Ωε|v|p-2v(v-hWε-uε)dx=Ω|v|p-2v(v-u)dx.

The aforementioned corrector term in the form hWε, where hW1,(Ωε,Ω) will be taken to satisfy the condition h(x)=H(v(x)) for a.e. xΩ, with H given by (2.4). These conditions rise naturally so that the term I2,ε above cancels out with the reaction term.

Remark 2.10.

In general, it is expected that the convergence u~εu can be improved to strong convergence by adding a corrector term. In fact, if σ is smooth, it is known that uε-H(uε)Wεu strongly in W01,p(Ω) (see, e.g., [37]). It is possible to adapt these arguments to the case of some maximal monotone graphs as, for instance, the one given by the Signorini boundary condition (see [12]).

3 Existence of uε and characterization of the function H

Proof of Proposition 2.2.

Consider the Banach space V=W01,p(Ωε,Ω). Let Aε:VV be the operator defined by

Aεv,w=Ωε|v|p-2vwdxfor any wW01,p(Ωε,Ω).

Then A is a hemicontinuous strictly monotone coercive operator [26]. Define φε:W01,p(Ωε,Ω)(-,+] by

φε(u)={ε-γSεΨ(trSε(u))𝑑Sif trSε(u(x))D(Ψ) for a.e. xSε,+otherwise.

It is clear that φε is a convex lower semicontinuous function with φε+. Since fV, we have that uε is a weak solution of problem (1.1) if and only if

Aε(uε)-f,v-uε+φε(v)-φε(uε)0for all vV.

Thus, the existence and uniqueness of a weak solution uε of problem (1.1) is consequence of [26, Chapter 2, Theorem 8.5].

In order to prove the a priori bound (2.3), let vW01,p(Ωε,Ω). Then, we have

Ωε|uε|pdx+ε-γSεΨ(uε)dSΩε|uε|p-2uεvdx+ε-γSεΨ(v)dS-Ωεf(v-uε)dx.

Given δ(0,1), we apply Young’s inequality, abδ|a|p+Cδ|b|p, to get

Ωε|uε|p-2uεvdxδΩε|uε|pdx+CδΩε|v|pdx.

Therefore, since Ψ0, taking v=0 and applying Hölder’s and Poincaré’s inequalities, we have

(1-δ)uεLp(Ωε)p+ε-γΨ(uε)L1(Sε)Ωεfuε𝑑xCfLp(Ω)uεLp(Ωε),

which leads to the result. ∎

Proof of Lemma 2.3.

Let Θn,p(s)=0|s|p-2s for s. Since σ-1 is also a maximal monotone graph of 2, for any p>1 and 0>0, the graph σ-1Θn,p is also a maximal monotone graph of 2. Indeed, let D(σ-1)=[a,b] for some -a<b+, and let (σ-1)0 be the principal section (i.e., the nondecreasing function) of the graph σ-1. This means that

(σ-1)0(r)=infσ-1(r),r[a,b].

Then, since Θn,p is strictly increasing, σ-1Θn,p is a monotone graph,

D(σ-1Θn,p)=[Θn,p-1(a),Θn,p-1(b)]and(σ-1Θn,p)0=(σ-1)0Θn,p.

In particular, if σ-1 is multivalued in some point c(a,b), then σ-1Θn,p(c) is the full interval

σ-1Θn,p(c)=[(σ-1)0(Θn,p(c)-),(σ-1)0(Θn,p(c)+)],

and this implies that σ-1Θn,p is a maximal monotone graph of 2 (see [2, Example 2.8.1]).

Now, since σ-1Θn,p is also a maximal monotone graph of 2, we know that (I+σ-1Θn,p) is an injective application such that R(I+σ-1Θn,p)= (see [2]). Thus, if H is defined by (1.5), then H is a nonexpansion on (see [2, Proposition 2.2]). Hence,

(I+σ-1Θn,p)(H(r))=r

for any r and, in consequence,

H(r)+σ-1Θn,p(H(r))=r.

In other words,

σ-1Θn,p(H(r))=r-H(r).

This implies that r-H(r)D(σ) for any r and that Θn,p(H(r))σ(r-H(r)) for any r, which proves that H(r) satisfies relation (2.4). Moreover, from the definition of H, it is obvious that H is nondecreasing (in fact if σ is strictly increasing, then H is also a strictly increasing function).

On the other hand, such function H(r) is the unique function satisfying relation (2.4), since applying the inverse graph

σ-1Θn,pH(I-H)

implies that (I+σ-1Θn,p)H=I, and so, necessarily, H=(I+σ-1Θn,p)-1. Of course, from the implicit formula, H is strictly increasing. ∎

4 Proof of Theorem 2.4

Since G0 is ball, it is easy to see that

wεj(x)=|x-Pεj|-n-pp-1-(ε/4)-n-pp-1(C0εα)-n-pp-1-(ε/4)-n-pp-1,xTεj,Gεj¯,(4.1)

is the unique solution of (2.8).

Lemma 4.1.

If Wε is defined by (2.9), then the following estimate holds:

Ωε|Wε|qdxKεn(p-q)n-pfor any 1qp.(4.2)

In particular,

Wε0in W01,p(Ω) as ε0.

Proof.

Estimate (4.2) is an explicit computation. For q=p, we obtain from it that, up to a subsequence, there exists W0W01,p(Ω) such that WεW0 in W01,p(Ω). For q<p, we have that Wε0 in W01,q(Ω), hence W0=0. ∎

Proof of Theorem 2.4.

Let v𝒞c(Ω) and h=H(v), with H: given by (1.5). Then hW1,(Ω). Let vε=v-hWεW01,p(Ωε,Ω), with WεW01,(Ω) defined by (2.9). Due to (2.7), we know that uε satisfies the inequality

Ωε|vε|p-2vε(vε-uε)dx+ε-γSε(Ψ(vε)-Ψ(uε))dSΩεf(vε-uε)dx.

Since Wε0 in Lp(Ω) (due to the compact inclusion), by Theorem 2.9, we can deduce that

limε0[I1,ε+I2,ε+I3,ε+ε-γSε(Ψ(vε)-Ψ(uε))𝑑S]limε0Ωεf(vε-uε)𝑑x=Ωf(v-u)𝑑x.

Since H: satisfies (2.4), by applying that if ξΨ(s0)=σ(s0), then Ψ(s)-Ψ(s0)ξ(s-s0), we can write

I2,ε+ε-γSε(Ψ(vε)-Ψ(uε))𝑑S=ε-γSε[Ψ(v-H(v))-Ψ(uε)-0|H(v)|p-2H(v)(v-H(v)-uε)]𝑑S0,

since 0|H(v(x))|p-2H(v(x))σ(v(x)-H(v(x))) for any xΩ¯. We can pass also to the limit in (2.10) and (2.11) to get that

Ω|v|p-2v(v-u)dx+Ω0|H(v)|p-2H(v)(v-u)dxΩf(v-u)dx,

and since v𝒞c(Ω) is arbitrary, by density, this also holds for every vW01,p(Ω). Hence, we get that u is the unique weak solution of (1.3). ∎

5 Proof of Theorem 2.9

The proof of Theorem 2.9 for p=2 can be found in [36], and for 2<p<n in [34]. Here we will complete the proof for 1<p<2. We need some auxiliary results.

Lemma 5.1 ([12]).

Let 1<p<2. Then there exists positive constant C=C(p) such that the inequality

||𝐚-𝐛|p-2(𝐚-𝐛)-(|𝐚|p-2𝐚-|𝐛|p-2𝐛)|C(|𝐚||𝐛|)p-12

is valid for all a,bRn.

By using this result, we prove the following lemma.

Lemma 5.2.

Let 1<p<2, n3, vW01,(Ω) and φW01,p(Ω). Let ηεW1,p(Ω) be such that ηεLq(Ω)0 for some q[1,p) as ε0. Then

limε0(Ωε|(v-ηε)|p-2(v-ηε)φdx)=limε0(Ωε|v|p-2vφdx-Ωε|ηε|p-2ηεφdx).

Proof.

By Lemma 5.1, by applying Hölder’s inequality, we have

|Ωε|(v-ηε)|p-2(v-ηε)φdx-(v|p-2v-|ηε|p-2ηε)φdx|CΩε|v|p-12|ηε|p-12|φ|dxKvp-12ηεLp+12(Ωε)p-12φLp+12(Ωε),

since 1<p+12<p. This proves the result. ∎

We have all the tools we need for the proof of Theorem 2.9.

Proof of Theorem 2.9.

As said before, it is enough to consider the case p(1,2). Applying Lemma 5.2, we obtain

limε0(Ωε|vε|p-2vε(vε-uε)dx)=limε0(J1,ε+J2,ε),

where

J1,ε=Ωε|v|p-2v(v-hWε-uε)dx,J2,ε=Ωε|(hWε)|p-2(hWε)(v-hWε-uε)dx.

Moreover,

limε0J1,ε=limε0(I1,ε+Ωε|v|p-2v(hWε)dx)=limε0I1,ε.

On the other hand,

limε0J2,ε=limε0(Ωε|Wε|p-2Wε(v-hWε-uε)dx)=limε0(jΥεTεj|wεj|p-2νwεj|h|p-2h(v-uε)dS+jΥεGεj|wεj|p-2νwεj|h|p-2h(v-h-uε)dS),

where νg is the usual normal derivative of g. Using (4.1), we get

νwεj|Tεj=ddrwεj|r=ε/4=-(n-p)22n-2p-1C0n-pp-1ε1p-1(p-1)(1-(C0εα)n-pp-1ε-n-pp-122n-2pp-1),νwεj|Gεj=-ddrwεj|r=aε=(n-p)ε-nn-p(p-1)C0(1-(C0εα)n-pp-1ε-n-pp-122n-2pp-1).

Therefore,

limε0J2,ε=limε0(AεεjΥεTεj|h|p-2h(v-uε)ds-ε-γSε((n-pp-1)p-1C01-p|h|p-2h)(v-h-uε)ds-Qε),

where

Aε=(n-pp-1)p-122n-2C0n-p(1-(C0εα)n-pp-1ε-n-pp-122n-2pp-1)p-1(5.1)

and

Qε=1-(1-aεn-pp-1εp-np-122n-2pp-1)p-1C0p-1(1-aεn-pp-1εp-np-122n-2pp-1)p-1(n-pp-1)p-1ε-γSε|h|p-2h(v-h-uε)dS.

It is an easy (but tedious) task to check that

limε0Qε=0,

which concludes the proof. ∎

6 Noncritical case and pointwise comparison of homogenized solutions with the critical case

For Am, let 𝒞(A) denote the space of continuous functions on A.

Theorem 6.1.

Let n3, p[2,n), 1<α<nn-p, fL(Ω) and rC(Ω¯). Let also σC(R) be nondecreasing such that σ(0)=0 and let uε be the solution of

{-Δpuε=f,xΩε,νpuε+ε-γσ(uε)=ε-γr,xSε,uε=0,xΩ.(6.1)

Then u~εunon-crit in W01,p(Ω), where unon-crit is the solution of

{-Δpu+𝒜^σ(u)=f+𝒜^rin Ω,u=0on Ω,(6.2)

with A^=C0n-1|G0|.

Proof.

Assume first that

0<k1σk2.

Then the result holds by [34, Theorem 3].

Applying the estimates in [29], we check that (Pεuε) is bounded in W01,p(Ω), hence there exists a limit u^ such that, up to a subsequence, Pεuεu^ strongly in Lp(Ω) and weakly in W01,p(Ω).

Let M be such that uεL(Ωε)M (see [8]). Let σδ be a sequence such that 0<k1,δσδk2,δ and σδσ in 𝒞([-M,M]) as δ0. Let uε,δ be the solution of (6.1) with σδ. We can check, again from estimates in [29], that

uε-uε,δLp(Ωε)Cσ-σδ𝒞([-M,M]).

Passing to the limit as ε0, indicating that Pεuε,δuδ in W01,p(Ω), where uδ is the solution of (6.2) with σδ, we have that

u^-uδLp(Ω)Cσ-σδ𝒞([-M,M]).

It is easy to check that uδu in Lp(Ω), where u is the solution of the problem with σ. Therefore, Pεuεu in Lp(Ω) as ε0 and u=u^. ∎

Theorem 6.2.

Let n3, p[2,n), α=nn-p, fL(Ω) and rC(Ω¯). Let also σC(R) be nondecreasing such that σ(0)=0 and let uε be the solution of (6.1). Then u~εucrit in W01,p(Ω), where ucrit is the solution of

{-Δpu+𝒜|H(x,u)|p-2H(x,u)=fin Ω,u=0on Ω,

and H is the solution of

0|H(x,s)|p-2H(x,s)=σ(s-H(x,s))-r(x)a.e. in Ω.

Sketch of proof.

We can apply the same reasoning as before and the fact that HδH, in the sense of maximal monotone graphs, as σδσ in 𝒞([-M,M]). ∎

Theorem 6.3.

Assume the conditions of the two previous theorems, f=0 and r(x)g(1)=1. Then, we have that ucritunon-crit.

Proof.

The condition on f and r guarantee that 0u1 in both cases. It is easy to check that H is increasing, and H(s)0 for s[0,1]. It is easy to establish the following inequality on the zero order terms:

0|H(s)|p-2H(s)𝒜^(σ(s)-g(1)).

Therefore, applying the comparison principle (see, e.g., [8]), we have the result. ∎

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About the article

Received: 2017-06-14

Accepted: 2017-06-16

Published Online: 2017-08-03


Funding Source: Ministerio de Educación, Cultura y Deporte

Award identifier / Grant number: FPU14/03702

The research of the first two authors was partially supported by the project ref. MTM2014-57113-P of the DGISPI (Spain) and as members of the Research Group MOMAT (Ref. 910480) of the UCM . The research of D. Gómez-Castro was supported by an FPU Grant from the Ministerio de Educación, Cultura y Deporte (Spain) (Ref. FPU14/03702).


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 679–693, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0140.

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