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

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 on ε . The generalized Sobolev space is constructed, and the epiconvergence method is considered to ﬁ nd the limit problem with interface conditions.


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 myo-cardial 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 ⊂ ℝ 3 , with a Lipschitz boundary ∂Ω, composed on a nanolayer B ε , with oscillating border Σ ± ε , of middle interface Σ, where B ε = fx ∈ Ω : jx 3 j ≤ εφðx ′ /εÞg and x = ðx ′ , x 3 Þ, Ω ε = Ω \ B ε , ε be a positive small enough parameter and φ be a0, 1½×0, 1½− periodic function. Let us consider a thermal problem on the body occupying the domain Ω, where a very high heat conductivity on the layer B ε 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 the conductivity is expressed by 1/ε α and the unknown u ε be the temperature, n be the outward normal to ∑ ± ε , pð:Þ > 1, α > 0, f ∈ L ∞ ðΩÞ, we denote by ½v the jump of the function v, Δ pð:Þ is pð:Þ-Laplace operator defined on sobolev space W 1,pð⋅Þ 0 ðΩÞ to its dual space W −1,pð⋅Þ ′ 0 ðΩÞ. 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 φ ε is a bounded real function, 0, ε½ 2 -periodic.
This paper is organized in the following way. In Section 2, we give some notations and assumptions on the function pð:Þ , 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.

Notations and Assumptions
In this section, we will give the notations that will be used throughout this paper: (iv) C will denote any constant with respect to ε and 0:+ ∞ = 0 Now, we assume that the function pð:Þ and r satisfying the following conditions there exists a continuous ω : (1). Note that problem (1) is equivalent to the minimization problem

Study of the Problem
F So, problems (1) and (5)  ðΩÞ. Next, we will be interested to study the 2 Journal of Function Spaces minimization problem (5), and the existence of its weak solutions is given in the following proposition.
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 solution u ε 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 ∇u ε . (2), (3), and for f ∈ L ∞ ðΩÞ. Then, ðu ε Þ ε>0 satisfies to
Proof. Since u ε satisfies to So, we have According to Hölder and Friedrich's inequalities, we obtain From (A.3) and Young inequality, we have From (11), we have ð from (3), we have pðxÞ ≤ p + in Ω, so that Therefore, we will have the assertions (8), (9), and from (A.3), we deduce that (u ε ) is bounded in W 1,pð:Þ 0 ðΩÞ. ☐ 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 of α, to obtain some information about the behavior of solution ðu ε Þ ε of problem (5) when ε close to zero, for this, we need to define an operator which transforms the functions defined on B ε to the functions defined on Σ, like in [15], let us define the following operator m ε : W 1,pð:Þ ðB ε Þ ⟶ W 1,pð:Þ ðΣÞ, by This mean operator satisfies the following results: The operator m ε definite by (17)   ðΩÞ which satisfies (8) and (9), Moreover, m ε u ε possesses a bounded subsequence in L pð:Þ ðΣÞ.

Proposition 5. The solution of the problem
Proof. Given that the solution of problem (5) satisfies Lemma 2 and according to the same lemma, then, the sequence u ε is bounded in W 1,pð:Þ 0 ðΩÞ, it follows that there exists an element u * ∈ W 1,pð:Þ 0 ðΩÞ and a subsequence of u ε , still denoted by u ε such that u ε ⇀ u * in W 1,pð:Þ 0 ðΩÞ. 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 and according to (9) For α = 1, according to the evaluation (19), the sequence ∇′m ε u ε possesses a subsequence, still denoted by ∇′m ε u ε weakly convergent to an element u 2 in L pð:Þ ðΣÞ 2 , as m ε u ε ⇀ u * | Σ in L pð:Þ ðΣÞ, so one concludes that m ε u ε ⇀ u * | Σ in W 1,pð:Þ ðΣÞ and ∇ ′ u * | Σ = u 2 . Hence, u * | Σ ∈ W 1,pð:Þ ðΣÞ. For α > 1, one shows, as in the case α = 1 and taking u 2 = 0, that u * | Σ = C. ☐ The limit behavior of the problem (5) will be derived with the epiconvergence method (see Definition B.1). (5). In this subsection, we will interest, according to the values of α, to find the limit problem of the problem (5). Consider the following energy functional given by

Limit Behavior of the Problem
We denote by τ f the weak topology on W 1,pð:Þ ðΩÞ and let The set V C ðΣÞ is a Banach space with the norm of W 1,pð:Þ 0 ðΩÞ. We show easily that VðΣÞ is a Banach space with the norm Let It is known that D α = G α .

Journal of Function Spaces
If α ≥ 1: Proof.

Journal of Function Spaces
In the sequel, one is interested to limit problem determination partner to the problem (5) Proof. Thanks to Lemma 2, the family (u ε ) is bounded in W Since for α > 1, F α equals+∞ on W 1,pð:Þ 0 According to the uniqueness of solutions of the problem (8), so that u ε admits an unique τ f -cluster point u * , and therefore, u ε ⇀ u * in W 1,pð:Þ 0 ðΩÞ. ☐ 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 thickness ε, 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 inter-face Σ, one has, on the interface Σ (the heat continuity), a bidimensional problem or the constant heat.

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 of ε. 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.
Theorem B.3 (see [18], Theorem 1.10). Suppose that (1) F ε admits a minimizer on X (2) The sequence ð u ε Þ is τ-relatively compact (3) The sequence F ε epiconverges to F in this topology τ Then, every cluster point u of the sequence ð u ε Þ minimizes F on X and if ð u ε ′ Þ ε ′ denotes the subsequence of ð u ε Þ ε which converges to u .

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.