# On Weak Solvability of Boundary Value Problem with Variable Exponent Arising in Nanostructure

Authors: Hicham Maadan; Nour Eddine Askour; Jamal Messaho
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This work is devoted to study the limit behavior of weak solutions of an elliptic problem with variable exponent, in a containing structure, of an oscillating nanolayer of thickness and periodicity parameter depending onε. The generalized Sobolev space is constructed, and the epiconvergence method is considered to find the limit problem with interface conditions.


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## Body

## 1. Introduction

Nanotechnology is the science that deals with matter at the scale of one billionth of a meter and is also the study of manipulating matter at the atomic and molecular scale. The development of nanotechnology requires the collaboration of all of the sciences. At the nanolevel, the physical properties of materials such as magnetic, electric, elastic, and thermal properties are fundamentally different from those at the macroscale, and it is important to appreciate at the outset that all such properties depend on scale as well as on the material. A nanoparticle is the most fundamental component in the fabrication of a nanostructure. Its size spans the range between 1 and 100 nm. Metallic nanoparticles have different physical and chemical properties from bulk metals (e.g., lower melting points, higher specific surface areas, specific optical properties, mechanical strengths, and specific magnetizations), properties that might prove attractive in various industrial applications. Nanostructures can also provide solutions to technological and environmental challenges in the areas of catalysis, biology, water treatment, energy conversion, and medicine. In biophysics, a mathematical anatomy of the cardiac left ventricle was considered by Koshelev et al. in [1], where the ventricle is composed of surfaces that model myocardial layers, each layer is filled with curves corresponding to myocardial fibers.Since superconductivity transition temperature rises so that particle diameter is small (less than 1 nm), it can be used to make high-temperature superconductivity material. Our aim is to study the composition of interconnected constitute parts in the nanoscale region where certain properties are amplified, and news behaviors and fundamental function of systems appear. Hence, there are three important effects which distinguish nanobehavior, and which do not occur in the macroenvironment; there are interfacial or surface effects, the effects of scale, and quantum effects due to a changed electronic structure. All three effects are interdependent and consequences of the extremely small size. The first plays a stone role in the kinetic theory of gases. In fact, nanoparticles are not inert in the host material, and that there is some interaction between the nanoparticles and the surrounding material, which is often modeled by a layer, which surrounds the nanoparticles but having different physical properties. Furthermore, to successfully develop nanotechnology, the major challenges are to properly understand material properties and also to be able to correctly predict their behavior when they interact with the system environment. Hence, we discuss mathematical models for problems of the enhanced thermal conductivity of nanofluids where potential mechanisms of enhanced heat transport in nanofluids are considered like high heat transport in the nanoparticles, since their thermal conductivity increases in a nonlinear way because most of their atoms are on the surface, and liquid nanolayer at the particle surface, which has a higher thermal conductivity than the liquid itself and so on, see for example [2].To do so, let us consider a body which occupies a bounded three-dimensional domain,Ω⊂ℝ3, with a Lipschitz boundary ∂Ω, composed on a nanolayer Bε, with oscillating border Σε±, of middle interface Σ, where Bε=x∈Ω:x3≤εφx′/εandx=x′,x3, Ωε=Ω\Bε, ε be a positive small enough parameter and φ be a0,1×0,1− periodic function. Let us consider a thermal problem on the body occupying the domain Ω, where a very high heat conductivity on the layer Bε is considered. The last body mentioned is subject to an outside temperature f, f:Ω⟶ℝ, and cooled at the boundary ∂Ω. The problem is modeled with the following equations(1)Δp⋅uε+f=0inΩε,1εαΔp⋅uε+f=0inBε,uε=0on∑ε±,∇uεp⋅−2∂uε∂nΩε=1εα∇uεp⋅−2∂uε∂nBεon∑ε±,uε=0on∂Ω,the conductivity is expressed by 1/εα and the unknown uε be the temperature, n be the outward normal to ∑ε±, p.>1,α>0, f∈L∞Ω, we denote by v the jump of the function v, Δp. is p.-Laplace operator defined on sobolev space W01,p⋅Ω to its dual space W0−1,p⋅′Ω. In setting of the constant exponent case, the asymptotic analysis of linear and nonlinear either in thermal and thermoelasticity problems with heat conductivity is widely treated by many authors, for example, in a structure with plate case, we can refer the reader to Ait Moussa and Licht in [3], Bourgeat and Tapiero in [4], Suquet and Sawczuck in [5], Brillard et al. in [6, 7], Pesqueira et al. in [8, 9], Silva et al. in [10, 11], Arrieta and Pereira in [12], and Sanchez-Palencia et al. in [13, 14], in other hand for a structure with an oscillating layer, Messaho et al. have studied the problem with very high conductivity in the layer in [15, 16] and Ameziane et al. have treated the case of a porous media behavior in [17]. Now, the aim of the present work is to study the problem of existence and limit behavior of a weak solutions for elliptic problem with variable exponent expressed by (1), which presents several mathematical difficulties either in terms of choosing a suitable Luxemburg norm for the considered Sobolev space or using the epiconvergence method to find the limit problem of our model. Accordingly, the limit behavior consist, via the epiconvergence method (see for instance [18, 19]), to find a minimization limit problem linked to the minimization problem related to (1). Where ε being a positive parameter intended to tend toward zero and φε is a bounded real function, 0,ε2-periodic.This paper is organized in the following way. In Section2, we give some notations and assumptions on the function p., functional spaces for our study, and results for the epiconvergence notion (in appendix) that will be used throughout this paper. We study problem (5) and its limit asymptotic in Section 5.


## 2. Notations and Assumptions

In this section, we will give the notations that will be used throughout this paper:(i)
Ω be a bounded domain in ℝ3 with measΩ>0, x=x′,x3 where x′=x1,x2(ii)
Y=0,1×0,1, φ:ℝ2⟶a1,a2, where φ is Y-periodic and a2≥a1>0(iii)
φεx′=φx′/ε, mφ=1/measY∫Yφx′dx′,ηα=limε⟶0ε1−α with α≥0(iv)
Cwilldenoteanyconstantwithrespecttoε and 0.+∞=0Now, we assume that the functionp. and r satisfying the following conditions(2)p.∈C0Ω¯withp.>1inΩ¯,(3)thereexistsp−,p+∈1,3suchthatp−=infx∈Ω¯pxandp+=supx∈Ω¯px,(4)thereexistsacontinuousω:ℝ+⟶ℝ+,∀x,y∈Ω,px−py≤ωx−ywithlimτ⟶0¯ωτlog1τ=0.


## 3. Main Results

### 3.1. Study of the Problem (1)

Note that problem (1) is equivalent to the minimization problem(5)infv∈W01,p.ΩΦεv+Fv,(6)Φεv=∫Ωε1px∇vpxdx+1εα∫Bε1px∇vpxdx,(7)Fv=−∫Ωf.vdx.So, problems (1) and (5) have the same weak solutions in W01,p.Ω. Next, we will be interested to study the minimization problem (5), and the existence of its weak solutions is given in the following proposition.Proposition 1.
Assume that assumptions (2) and (3) hold. If f∈L∞Ω, then, the problem (5) admits an unique solution uε in W01,p.Ω.The proof of this proposition is based on classical convexity arguments see for example [20, 21].In the sequel, we focus on the limit behavior of the solutionuε of problem (5) with respect to the values of α. In order to establish this behavior, we use the epiconvergence method (see Definition B.1), to starting this, in one hand, we need to determine the space and its topology in applying the epiconvergence method. Now, we give in the following lemma, the estimations on ∇uε.Lemma 2.
Assuming that there exists constantsC>0 and a sequence uεε>0⊂W01,p.Ω such that Φεuε+Fuε≤C, under (2), (3), and for f∈L∞Ω. Then, uεε>0 satisfies to(8)∫Ωε∇uεpxdx≤C,(9)∫Bε∇uεpxdx≤Cεα,
Moreover,uε is bounded in W01,p.Ω.Proof.
Sinceuε satisfies to(10)∫Ω1px∇uεpxdx+1εα∫Bε1px∇uεpxdx−∫Ωfuεdx≤C.
So, we have(11)∫Ω1px∇uεpxdx+1εα∫Bε1px∇uεpxdx≤C+∫Ωfuεdx.
According to Hölder and Friedrich’s inequalities, we obtain(12)∫Ωfuεdx≤2fLp.ΩuεLp.Ω≤C∇uεLp.Ω.
From (A.3) and Young inequality, we have(13)∫Ωfuε≤CmaxAΩ,p.∇uε1/p−,AΩ,p.∇uε1/p+≤C+max1p−AΩ,p.∇uε,1p+AΩ,p.∇uε≤C+1p−AΩ,p.∇uε≤C+1p−∫Ωε∇uεpxdx+1p−∫Bε∇uεpxdx≤C+1p−∫Ωε∇uεpxdx+1εαp−∫Bε∇uεpxdx.
From (11), we have(14)∫Ωε1px∇uεpxdx+1εα∫Bε1px∇uεpxdx≤C+1p−∫Ωε∇uεpxdx+1εαp−∫Bε∇uεpxdx.
So(15)1p−′∫Ωε1px∇uεpxdx+1p−′1εα∫Bε1px∇uεpxdx≤C,from (3), we have px≤p+ in Ω, so that(16)1p+p−′∫Ωε∇uεpxdx+1p+p−′1εα∫Bε∇uεpxdx≤C.
Therefore, we will have the assertions (8), (9), and from (A.3), we deduce that (uε) is bounded in W01,p.Ω.☐As a consequence, we can confirm that the solution of problem (5) satisfies Lemma 2.Now, in the following proposition, we will discuss, according to the real values ofα, to obtain some information about the behavior of solution uεε of problem (5) when ε close to zero, for this, we need to define an operator which transforms the functions defined on Bε to the functions defined on Σ, like in [15], let us define the following operator mε:W1,p.Bε⟶W1,p.Σ, by(17)mεux′=12εφεx′∫−εφεx′εφεx′ux′,x3dx3,∀u∈W1,p.Bε.This mean operator satisfies the following results:Lemma 3.
The operatormε definite by (17) is linear and bounded of Lp.Bε (respectively, W1,p.Bε) in Lp.Σ (respectively, W1,p.Σ), with norm ≤Cε−1/p−, moreover, for all u∈W1,p.Bε, we have(18)∫Σmεu−u∣Σpx≤Cεp−−1∫Bε∇upx.Lemma 4.
Letuεε>0⊂W01,p.Ω which satisfies (8) and (9), then(19)∫Σ∇′mεuεpx≤Cεα−1.
Moreover,mεuε possesses a bounded subsequence in Lp.Σ.The proof of these two previous lemmas is similar to the lemmas 4.2 and 4.3 in the reference [15].Proposition 5.
The solution of the problem (5), uεε, possesses a subsequence denoted also uεε weakly convergent toward an element u∗ in W01,p.Ω satisfying(20)u∗∣Σ∈Lp.Σ,ifα=1,u∗∣Σ∈W1,p.Σ,∀α>1,u∗∣Σ=C.Proof.
Given that the solution of problem (5) satisfies Lemma 2 and according to the same lemma, then, the sequence uε is bounded in W01,p.Ω, it follows that there exists an element u∗∈W01,p.Ω and a subsequence of uε, still denoted by uε such that uε⇀u∗ in W01,p.Ω. Under (2), (3), and (4), the trace operator proprieties holds in the Sobolev space with variable exponent, for more details, see for example [22], so(21)uε∣Σ⇀u∗∣ΣinLp.Σ,thanks to Lemma 3, we have(22)∫Σmεuε−uε∣Σpx≤Cεp−−1∫Bε∇uεpx,and according to (9), we have(23)∫Σmεuε−uε∣Σpx≤Cεα+p−−1.
Forα=1, according to the evaluation (19), the sequence ∇′mεuε possesses a subsequence, still denoted by ∇′mεuε weakly convergent to an element u2 in Lp.Σ2, as mεuε⇀u∗∣ΣinLp.Σ, so one concludes that mεuε⇀u∗∣ΣinW1,p.Σ and ∇′u∗∣Σ=u2. Hence, u∗∣Σ∈W1,p.Σ.
Forα>1, one shows, as in the case α=1 and taking u2=0, that u∗∣Σ=C.☐The limit behavior of the problem (5) will be derived with the epiconvergence method (see Definition B.1).


### 3.2. Limit Behavior of the Problem (5)

In this subsection, we will interest, according to the values ofα, to find the limit problem of the problem (5). Consider the following energy functional given by(24)Fεu=∫Ωε1px∇upx+1εα∫Bε1px∇upx,∀u∈W1,p.Ω,(25)Φu=−∫Ωfu,∀u∈W1,p.Ω.We denote byτf the weak topology on W1,p.Ω and let(26)VΣ=u∈W01,p.Ω:uΣ∈W1,p.Σ,(27)VCΣ=u∈W01,p.Ω:uΣ=C.The setVCΣ is a Banach space with the norm of W01,p.Ω. We show easily that VΣ is a Banach space with the norm(28)u↦∇uLp.Ω3+∇′u∣ΣLpxΣ2.Let(29)Gα=u∈W01,p.Ω:ηαuΣ∈W1,p.Σifα≤1,VCifα>1.(30)Dα=DΩifα≤1,u∈DΩ:uΣ=Cifα>1.It is known thatDα¯=Gα.Theorem 6.
According to the values ofα, there exists a functional Fα defined on W01,p.Ω with value in ℝ∪+∞ such that τf−limeFε=FαinW01,pΩ, where the functional Fα is given as follows
If0≤α<1:(31)Fαu=∫Ω1px∇upx,∀u∈W01,p.Ω.
Ifα≥1:(32)Fαu=∫Ω1px∇upx+2mφηα∫Σ1px∇′u∣Σpxifu∈Gα,+∞ifu∈W01,p.Ω\Gα.Proof.
(a)
One Is Going to Determine the Upper Epilimit. Let u∈Gα⊂W01,p.Ω, there exists a sequence un in Dα such that(33)un⟶uinGα,whenn⟶+∞.
So thatun⟶u in W01,p.Ω. Let θ be a smooth function satisfying(34)θt=1ift≤1,θt=0ift≥2andθ′t≤2∀t∈ℝ,and set(35)θεx=θx3εφεx′;we define(36)uε,n=θεxun∣Σ+1−θεxun,
One shows easily thatuε,n∈W01,p.Ω and uε,n⟶un in Gα, when ε⟶0. Since(37)Fεuε,n=∫Ωε1px∇uε,npx+1εα∫Bε1px∇uε,npx,so that(38)Fεuε,n=∫x3>2εφε1px∇uε,npx+∫εφε<x3<2εφε1px∇uε,npx+1εα∫Bε1px∇uε,npx=∫x3>2εφε1px∇unpx+∫εφε<x3<2εφε1px∇uε,npx+2ε1−α∫Σφε1px∇′un∣Σpx.
Sinceφε is bounded, one verifies easily that(39)limε⟶0∫εφε<x3<2εφε1px∇uε,npx=0.(1)
Ifα≤1. Since φε⇀∗mφ in L∞Σ and ε1−α⟶ηα, it follows that(40)limε⟶02ε1−α∫Σφε1px∇′un∣Σpx=2mφηα∫Σ1px∇′un∣Σpx.
By passing to the upper limit, one has(41)limsupε⟶0Fεuε,n=limsupε⟶0∫x3>2εφε1px∇unpx+2ε1−α∫Σφε1px∇′un∣Σpx=∫Ω1px∇unpx+2mφηα∫Σ1px∇′un∣Σpx.(2)
Ifα>1. By passing to the upper limit, one has(42)limsupε⟶0Fεuε,n=limsupε⟶0∫x3>2εφε1px∇unpx=∫Ω1px∇unpx.
Sinceun⟶u in Gα, when n⟶+∞. According to the classical result, diagonalization’s lemma ([18], Lemma 1.15), there exists a function nε:ℝ+⟶ℕ increasing to +∞ when ε⟶0, such that uε,nε⟶u in Gα when ε⟶0. While n approaches +∞, one will have(1)
Ifα≠1(43)limsupε⟶0Fεuε,nε≤limsupn⟶+∞limsupε⟶0Fεuε,n≤∫Ω1px∇upx.(2)
Ifα=1(44)limsupε⟶0Fεuε,nε≤limsupn⟶+∞limsupε⟶0Fεuε,n≤∫Ω1px∇upx+2mφηα∫Σ1px∇′u∣Σpx.
Ifu∈W01,p.Ω\Gα, it is clear that, for every uε∈W01,p.Ω, uε⇀u in W01,p.Ω, one has(45)limsupε⟶0Fεuε≤+∞.(b)
One is going to determine the Lower Epilimit. Let u∈Gα and uε be a sequence in W01,p.Ω such thatuε⇀u in W01,p.Ω, so that(46)χΩε∇uε⇀∇uinLp.Ω3.(1)
Ifα≠1. Since(47)Fεuε≥∫Ωε1px∇uεpx.
According to (5) and by passing to the lower limit, one obtains(48)liminfε⟶0Fεuε≥∫Ω1px∇upx.(2)
Ifα=1. If liminfε⟶0Fεuε=+∞, there is nothing to prove, because(49)∫Ω1px∇upx+2mφηα∫Σ1px∇′u∣Σpx≤+∞.
Otherwise,liminfε⟶0Fεuε<+∞, there exists a subsequence of Fεuε still denoted by Fεuε and a constant C>0, such that Fεuε≤C, which implies that(50)1εα∫Bε1px∇uεpx≤C.
Souε satisfies the hypothesis of the Lemma 4, and according to this last, ∇′mεuε is bounded in Lp.Σ2, so there exists an element u1∈Lp.Σ2 and a subsequence of ∇′mεuε, still denoted by ∇′mεuε, such that ∇′mεuε⇀u1 in Lp.Σ2, since uε∣Σ⇀u∣Σ in Lp.Σ, and thanks to (18) and (50), one has mεuε⇀u∣Σ in Lp.Σ, then mεuε⇀u∣Σ in W1,p.Σ, therefore, u1=∇′u∣Σ, so that ∇′mεuε⇀∇′u∣Σ in Lp.Σ2. One has(51)Fεuε≥∫Ωε1px∇uεpx+1εα∫Bε1px∇uεpx≥∫Ωε1px∇uεpx+2ε1−α∫Σφε1px∇′mεuεpx.
Using the subdifferential inequality of(52)v⟶2ε1−α∫Σφεpxvpx,∀v∈Lp.Σ2,one has(53)Fεuε≥∫Ωε1px∇uεpx+2ε1−α∫Σφε1px∇′u∣Σpx+2ε1−α∫Σφε∇′u∣Σpx−2∇′u∣Σ∇′mεuε−∇′u∣Σ.
Thanks to LemmaB.4, one has φε⟶mφ in Lp′.Σ, so according to (5) and by passing to the lower limit, one obtains(54)liminfε⟶0Fεuε≥∫Ω1px∇upx+2mφηα∫Σ1px∇′u∣Σpx.
Ifu∈W01,p.Ω\Gα and uε∈W01,p.Ω, such that uε⇀u in W01,p.Ω.
Assume that(55)liminfε⟶0Fεuε<+∞.
So there exists a constantC>0 and a subsequence of Fεuε, still denoted by Fεuε, such that(56)Fεuε<C.
For0≤α<1, there is nothing to prove.
Otherwise, one takes the same way as in the caseu∈Gα=1, one has ∇′mεuε is bounded in Lp.Σ2, so there exists an element u1∈Lp.Σ2 and a subsequence of ∇′mεuε, still denoted by ∇′mεuε, such that ∇′mεuε⇀u1 in Lp.Σ2, since uε∣Σ⇀u∣Σ in Lp.Σ, and thanks to (18) and (56), one has mεuε⇀u∣Σ in Lp.Σ, then mεuε⇀u∣Σ in W1,p.Σ, so that u∈Gα what contradicts the fact that u∈Gα, so that(57)liminfε⟶0Fεuε=+∞.
Hence, the proof of Theorem6 is complete.☐In the sequel, one is interested to limit problem determination partner to the problem (5), when ε approaches zero. Thanks to the epiconvergence results, (see Theorem B.3, Proposition B.2), Theorem 6, and according to τf-continuity of Φ in W01,p.Ω, one has Fε+Φτf−epilimit toward Fα+Φ in W01,p.Ω.Proposition 7.
For everyf∈L∞Ω and according to the parameter values of α, there exists u∗∈W01,p.Ω satisfying(58)uε⇀u∗inW01,p.Ω,(59)Fαu∗+Φu∗=infv∈GαFαv+Φv.Proof.
Thanks to Lemma2, the family (uε) is bounded in W01,p.Ω, therefore, it possess a τf− cluster point u∗ in W01,p.Ω. And thanks to a classical epiconvergence result (see Theorem B.3), one has u∗ is a solution of the problem find(60)infv∈W01,p.ΩFαv+Φv.
Since forα>1, Fα equals+∞ on W01,p.Ω\Gα, (60) becomes(61)infv∈GαFαv+Φv.
According to the uniqueness of solutions of the problem (8), so that uε admits an unique τf-cluster point u∗, and therefore, uε⇀u∗ in W01,p.Ω.☐Remark 8.
One shows that the limit behavior of a constituted structure of two mediums of constant conductivity united by an oscillating nonlinear nanolayer of thicknessε, which the conductivity depends on the negative powers of ε, is described by a problem with interface Σ (Σ the middle interface of the nanolayer). Following the powers of ε, to the interface Σ, one has, on the interface Σ (the heat continuity), a bidimensional problem or the constant heat.


## 3.1. Study of the Problem (1)

Note that problem (1) is equivalent to the minimization problem(5)infv∈W01,p.ΩΦεv+Fv,(6)Φεv=∫Ωε1px∇vpxdx+1εα∫Bε1px∇vpxdx,(7)Fv=−∫Ωf.vdx.So, problems (1) and (5) have the same weak solutions in W01,p.Ω. Next, we will be interested to study the minimization problem (5), and the existence of its weak solutions is given in the following proposition.Proposition 1.
Assume that assumptions (2) and (3) hold. If f∈L∞Ω, then, the problem (5) admits an unique solution uε in W01,p.Ω.The proof of this proposition is based on classical convexity arguments see for example [20, 21].In the sequel, we focus on the limit behavior of the solutionuε of problem (5) with respect to the values of α. In order to establish this behavior, we use the epiconvergence method (see Definition B.1), to starting this, in one hand, we need to determine the space and its topology in applying the epiconvergence method. Now, we give in the following lemma, the estimations on ∇uε.Lemma 2.
Assuming that there exists constantsC>0 and a sequence uεε>0⊂W01,p.Ω such that Φεuε+Fuε≤C, under (2), (3), and for f∈L∞Ω. Then, uεε>0 satisfies to(8)∫Ωε∇uεpxdx≤C,(9)∫Bε∇uεpxdx≤Cεα,
Moreover,uε is bounded in W01,p.Ω.Proof.
Sinceuε satisfies to(10)∫Ω1px∇uεpxdx+1εα∫Bε1px∇uεpxdx−∫Ωfuεdx≤C.
So, we have(11)∫Ω1px∇uεpxdx+1εα∫Bε1px∇uεpxdx≤C+∫Ωfuεdx.
According to Hölder and Friedrich’s inequalities, we obtain(12)∫Ωfuεdx≤2fLp.ΩuεLp.Ω≤C∇uεLp.Ω.
From (A.3) and Young inequality, we have(13)∫Ωfuε≤CmaxAΩ,p.∇uε1/p−,AΩ,p.∇uε1/p+≤C+max1p−AΩ,p.∇uε,1p+AΩ,p.∇uε≤C+1p−AΩ,p.∇uε≤C+1p−∫Ωε∇uεpxdx+1p−∫Bε∇uεpxdx≤C+1p−∫Ωε∇uεpxdx+1εαp−∫Bε∇uεpxdx.
From (11), we have(14)∫Ωε1px∇uεpxdx+1εα∫Bε1px∇uεpxdx≤C+1p−∫Ωε∇uεpxdx+1εαp−∫Bε∇uεpxdx.
So(15)1p−′∫Ωε1px∇uεpxdx+1p−′1εα∫Bε1px∇uεpxdx≤C,from (3), we have px≤p+ in Ω, so that(16)1p+p−′∫Ωε∇uεpxdx+1p+p−′1εα∫Bε∇uεpxdx≤C.
Therefore, we will have the assertions (8), (9), and from (A.3), we deduce that (uε) is bounded in W01,p.Ω.☐As a consequence, we can confirm that the solution of problem (5) satisfies Lemma 2.Now, in the following proposition, we will discuss, according to the real values ofα, to obtain some information about the behavior of solution uεε of problem (5) when ε close to zero, for this, we need to define an operator which transforms the functions defined on Bε to the functions defined on Σ, like in [15], let us define the following operator mε:W1,p.Bε⟶W1,p.Σ, by(17)mεux′=12εφεx′∫−εφεx′εφεx′ux′,x3dx3,∀u∈W1,p.Bε.This mean operator satisfies the following results:Lemma 3.
The operatormε definite by (17) is linear and bounded of Lp.Bε (respectively, W1,p.Bε) in Lp.Σ (respectively, W1,p.Σ), with norm ≤Cε−1/p−, moreover, for all u∈W1,p.Bε, we have(18)∫Σmεu−u∣Σpx≤Cεp−−1∫Bε∇upx.Lemma 4.
Letuεε>0⊂W01,p.Ω which satisfies (8) and (9), then(19)∫Σ∇′mεuεpx≤Cεα−1.
Moreover,mεuε possesses a bounded subsequence in Lp.Σ.The proof of these two previous lemmas is similar to the lemmas 4.2 and 4.3 in the reference [15].Proposition 5.
The solution of the problem (5), uεε, possesses a subsequence denoted also uεε weakly convergent toward an element u∗ in W01,p.Ω satisfying(20)u∗∣Σ∈Lp.Σ,ifα=1,u∗∣Σ∈W1,p.Σ,∀α>1,u∗∣Σ=C.Proof.
Given that the solution of problem (5) satisfies Lemma 2 and according to the same lemma, then, the sequence uε is bounded in W01,p.Ω, it follows that there exists an element u∗∈W01,p.Ω and a subsequence of uε, still denoted by uε such that uε⇀u∗ in W01,p.Ω. Under (2), (3), and (4), the trace operator proprieties holds in the Sobolev space with variable exponent, for more details, see for example [22], so(21)uε∣Σ⇀u∗∣ΣinLp.Σ,thanks to Lemma 3, we have(22)∫Σmεuε−uε∣Σpx≤Cεp−−1∫Bε∇uεpx,and according to (9), we have(23)∫Σmεuε−uε∣Σpx≤Cεα+p−−1.
Forα=1, according to the evaluation (19), the sequence ∇′mεuε possesses a subsequence, still denoted by ∇′mεuε weakly convergent to an element u2 in Lp.Σ2, as mεuε⇀u∗∣ΣinLp.Σ, so one concludes that mεuε⇀u∗∣ΣinW1,p.Σ and ∇′u∗∣Σ=u2. Hence, u∗∣Σ∈W1,p.Σ.
Forα>1, one shows, as in the case α=1 and taking u2=0, that u∗∣Σ=C.☐The limit behavior of the problem (5) will be derived with the epiconvergence method (see Definition B.1).


## 3.2. Limit Behavior of the Problem (5)

In this subsection, we will interest, according to the values ofα, to find the limit problem of the problem (5). Consider the following energy functional given by(24)Fεu=∫Ωε1px∇upx+1εα∫Bε1px∇upx,∀u∈W1,p.Ω,(25)Φu=−∫Ωfu,∀u∈W1,p.Ω.We denote byτf the weak topology on W1,p.Ω and let(26)VΣ=u∈W01,p.Ω:uΣ∈W1,p.Σ,(27)VCΣ=u∈W01,p.Ω:uΣ=C.The setVCΣ is a Banach space with the norm of W01,p.Ω. We show easily that VΣ is a Banach space with the norm(28)u↦∇uLp.Ω3+∇′u∣ΣLpxΣ2.Let(29)Gα=u∈W01,p.Ω:ηαuΣ∈W1,p.Σifα≤1,VCifα>1.(30)Dα=DΩifα≤1,u∈DΩ:uΣ=Cifα>1.It is known thatDα¯=Gα.Theorem 6.
According to the values ofα, there exists a functional Fα defined on W01,p.Ω with value in ℝ∪+∞ such that τf−limeFε=FαinW01,pΩ, where the functional Fα is given as follows
If0≤α<1:(31)Fαu=∫Ω1px∇upx,∀u∈W01,p.Ω.
Ifα≥1:(32)Fαu=∫Ω1px∇upx+2mφηα∫Σ1px∇′u∣Σpxifu∈Gα,+∞ifu∈W01,p.Ω\Gα.Proof.
(a)
One Is Going to Determine the Upper Epilimit. Let u∈Gα⊂W01,p.Ω, there exists a sequence un in Dα such that(33)un⟶uinGα,whenn⟶+∞.
So thatun⟶u in W01,p.Ω. Let θ be a smooth function satisfying(34)θt=1ift≤1,θt=0ift≥2andθ′t≤2∀t∈ℝ,and set(35)θεx=θx3εφεx′;we define(36)uε,n=θεxun∣Σ+1−θεxun,
One shows easily thatuε,n∈W01,p.Ω and uε,n⟶un in Gα, when ε⟶0. Since(37)Fεuε,n=∫Ωε1px∇uε,npx+1εα∫Bε1px∇uε,npx,so that(38)Fεuε,n=∫x3>2εφε1px∇uε,npx+∫εφε<x3<2εφε1px∇uε,npx+1εα∫Bε1px∇uε,npx=∫x3>2εφε1px∇unpx+∫εφε<x3<2εφε1px∇uε,npx+2ε1−α∫Σφε1px∇′un∣Σpx.
Sinceφε is bounded, one verifies easily that(39)limε⟶0∫εφε<x3<2εφε1px∇uε,npx=0.(1)
Ifα≤1. Since φε⇀∗mφ in L∞Σ and ε1−α⟶ηα, it follows that(40)limε⟶02ε1−α∫Σφε1px∇′un∣Σpx=2mφηα∫Σ1px∇′un∣Σpx.
By passing to the upper limit, one has(41)limsupε⟶0Fεuε,n=limsupε⟶0∫x3>2εφε1px∇unpx+2ε1−α∫Σφε1px∇′un∣Σpx=∫Ω1px∇unpx+2mφηα∫Σ1px∇′un∣Σpx.(2)
Ifα>1. By passing to the upper limit, one has(42)limsupε⟶0Fεuε,n=limsupε⟶0∫x3>2εφε1px∇unpx=∫Ω1px∇unpx.
Sinceun⟶u in Gα, when n⟶+∞. According to the classical result, diagonalization’s lemma ([18], Lemma 1.15), there exists a function nε:ℝ+⟶ℕ increasing to +∞ when ε⟶0, such that uε,nε⟶u in Gα when ε⟶0. While n approaches +∞, one will have(1)
Ifα≠1(43)limsupε⟶0Fεuε,nε≤limsupn⟶+∞limsupε⟶0Fεuε,n≤∫Ω1px∇upx.(2)
Ifα=1(44)limsupε⟶0Fεuε,nε≤limsupn⟶+∞limsupε⟶0Fεuε,n≤∫Ω1px∇upx+2mφηα∫Σ1px∇′u∣Σpx.
Ifu∈W01,p.Ω\Gα, it is clear that, for every uε∈W01,p.Ω, uε⇀u in W01,p.Ω, one has(45)limsupε⟶0Fεuε≤+∞.(b)
One is going to determine the Lower Epilimit. Let u∈Gα and uε be a sequence in W01,p.Ω such thatuε⇀u in W01,p.Ω, so that(46)χΩε∇uε⇀∇uinLp.Ω3.(1)
Ifα≠1. Since(47)Fεuε≥∫Ωε1px∇uεpx.
According to (5) and by passing to the lower limit, one obtains(48)liminfε⟶0Fεuε≥∫Ω1px∇upx.(2)
Ifα=1. If liminfε⟶0Fεuε=+∞, there is nothing to prove, because(49)∫Ω1px∇upx+2mφηα∫Σ1px∇′u∣Σpx≤+∞.
Otherwise,liminfε⟶0Fεuε<+∞, there exists a subsequence of Fεuε still denoted by Fεuε and a constant C>0, such that Fεuε≤C, which implies that(50)1εα∫Bε1px∇uεpx≤C.
Souε satisfies the hypothesis of the Lemma 4, and according to this last, ∇′mεuε is bounded in Lp.Σ2, so there exists an element u1∈Lp.Σ2 and a subsequence of ∇′mεuε, still denoted by ∇′mεuε, such that ∇′mεuε⇀u1 in Lp.Σ2, since uε∣Σ⇀u∣Σ in Lp.Σ, and thanks to (18) and (50), one has mεuε⇀u∣Σ in Lp.Σ, then mεuε⇀u∣Σ in W1,p.Σ, therefore, u1=∇′u∣Σ, so that ∇′mεuε⇀∇′u∣Σ in Lp.Σ2. One has(51)Fεuε≥∫Ωε1px∇uεpx+1εα∫Bε1px∇uεpx≥∫Ωε1px∇uεpx+2ε1−α∫Σφε1px∇′mεuεpx.
Using the subdifferential inequality of(52)v⟶2ε1−α∫Σφεpxvpx,∀v∈Lp.Σ2,one has(53)Fεuε≥∫Ωε1px∇uεpx+2ε1−α∫Σφε1px∇′u∣Σpx+2ε1−α∫Σφε∇′u∣Σpx−2∇′u∣Σ∇′mεuε−∇′u∣Σ.
Thanks to LemmaB.4, one has φε⟶mφ in Lp′.Σ, so according to (5) and by passing to the lower limit, one obtains(54)liminfε⟶0Fεuε≥∫Ω1px∇upx+2mφηα∫Σ1px∇′u∣Σpx.
Ifu∈W01,p.Ω\Gα and uε∈W01,p.Ω, such that uε⇀u in W01,p.Ω.
Assume that(55)liminfε⟶0Fεuε<+∞.
So there exists a constantC>0 and a subsequence of Fεuε, still denoted by Fεuε, such that(56)Fεuε<C.
For0≤α<1, there is nothing to prove.
Otherwise, one takes the same way as in the caseu∈Gα=1, one has ∇′mεuε is bounded in Lp.Σ2, so there exists an element u1∈Lp.Σ2 and a subsequence of ∇′mεuε, still denoted by ∇′mεuε, such that ∇′mεuε⇀u1 in Lp.Σ2, since uε∣Σ⇀u∣Σ in Lp.Σ, and thanks to (18) and (56), one has mεuε⇀u∣Σ in Lp.Σ, then mεuε⇀u∣Σ in W1,p.Σ, so that u∈Gα what contradicts the fact that u∈Gα, so that(57)liminfε⟶0Fεuε=+∞.
Hence, the proof of Theorem6 is complete.☐In the sequel, one is interested to limit problem determination partner to the problem (5), when ε approaches zero. Thanks to the epiconvergence results, (see Theorem B.3, Proposition B.2), Theorem 6, and according to τf-continuity of Φ in W01,p.Ω, one has Fε+Φτf−epilimit toward Fα+Φ in W01,p.Ω.Proposition 7.
For everyf∈L∞Ω and according to the parameter values of α, there exists u∗∈W01,p.Ω satisfying(58)uε⇀u∗inW01,p.Ω,(59)Fαu∗+Φu∗=infv∈GαFαv+Φv.Proof.
Thanks to Lemma2, the family (uε) is bounded in W01,p.Ω, therefore, it possess a τf− cluster point u∗ in W01,p.Ω. And thanks to a classical epiconvergence result (see Theorem B.3), one has u∗ is a solution of the problem find(60)infv∈W01,p.ΩFαv+Φv.
Since forα>1, Fα equals+∞ on W01,p.Ω\Gα, (60) becomes(61)infv∈GαFαv+Φv.
According to the uniqueness of solutions of the problem (8), so that uε admits an unique τf-cluster point u∗, and therefore, uε⇀u∗ in W01,p.Ω.☐Remark 8.
One shows that the limit behavior of a constituted structure of two mediums of constant conductivity united by an oscillating nonlinear nanolayer of thicknessε, which the conductivity depends on the negative powers of ε, is described by a problem with interface Σ (Σ the middle interface of the nanolayer). Following the powers of ε, to the interface Σ, one has, on the interface Σ (the heat continuity), a bidimensional problem or the constant heat.


## 4. Conclusion

In this study, we have found out the limit behavior of weak solutions of the problem with variable exponent, of a containing structure, an oscillating nanolayer of thickness, and periodicity parameter depending ofε. The epiconvergence method, which is extremely popular in the last years and which has been crucial for the developments of the calculus of variations and of the theory of partial differential equations, is considered to find the limit problems with interface conditions. The bounds and the behavior of the solution are discussed and the obtained results agree with the results obtained in some previous works where the exponent is constant in homogenization theory. In future works, we extend the proposed scheme elliptic problem with variable exponent with nonhomogenous interfaces conditions in a nanostructure.



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