Behavior of the 𝑝-Laplacian on Thin Domains

Abstract

We give the characterization of the limiting behavior of solutions of elliptic equations driven by the 𝑝-Laplacian operator with Neumann boundary conditions posed in a family of thin domains.

1. Introduction

The investigation of parabolic and elliptic equations on thin domains has received considerable attention over the last twenty years. Such equations can appear motivated by homogenization problems in thin structures as in [1–7], as well as in the parabolic counterpart, associated with the continuity of global attractors for dissipative equations as in [1, 8–15]. Whatever the motivations that appear, the key point in the study of any kind of perturbation problem is to find the limiting one. In this specific domain perturbation problem (thin domains), it means to find an equation posed in a lower dimensional domain in order to compare the perturbed problems with. Our contribution in this short note goes in this direction. We give the characterization of the limiting problem of a family of elliptic equations driven by the 𝑝-Laplacian operator. This can be used, for example, in the study of the asymptotic behavior (attractors) of dissipative equations governed by the 𝑝-Laplacian on thin domains, which is associated with localized large diffusion phenomena, see, for example, [16]. This is the first step in order to consider other aspects as the asymptotic dynamics (attractors). For the best of our knowledge this is an untouched topic in the literature and can be the starting point for investigation of quasi-linear parabolic equations on thin domains which is relevant in a variety of physical phenomena as non-Newtonian fluids as well as in flow through porous media.

In order to set up the problem, let 𝜔 be a smooth bounded domain in ℝ^𝑛, 𝑛≥1, and 𝑔∈𝐶²(𝜔;ℝ) a positive function; 𝜖 will represent a small positive parameter which will converge to zero. We consider the family of domains Ω^𝜖⊂ℝ^𝑛+1 defined by Ω^𝜖∶=𝑥,𝑦∈ℝ^𝑛×ℝ∶𝑥∈𝜔,0<𝑦<𝜖𝑔(𝑥). (1) The aim of this paper is to characterize the limiting problem (𝜖=0) for the family of elliptic equations −Δ_𝑝𝑢+|𝑢|^𝑝−2𝑢=𝑓^𝜖,inΩ^𝜖,𝜕𝑢𝜕𝜂^𝜖=0,on𝜕Ω^𝜖, (2) where 𝑝>2, 𝑓^𝜖∈𝐿^𝑞(Ω^𝜖), (1/𝑝)+(1/𝑞)=1, Δ_𝑝𝑢∶=div(|∇𝑢|^𝑝−2∇𝑢) denotes the 𝑝-Laplacian operator and 𝜂^𝜖 denotes the outward unitary normal vector field to 𝜕Ω^𝜖.

Definition 1. Given 𝑓^𝜖∈𝐿^𝑞(Ω^𝜖), 1<𝑞<2, one says that 𝑢∈𝑊^1,𝑝(Ω^𝜖), (1/𝑝)+(1/𝑞)=1 is a solution of (2) if _Ω𝜖|∇𝑢|^𝑝−2∇𝑢⋅∇𝜑+|𝑢|^𝑝−2𝑢𝜑𝑑𝑥𝑑𝑦=_Ω𝜖𝑓^𝜖𝜑𝑑𝑥𝑑𝑦, (3) for all 𝜑∈𝑊^1,𝑝(Ω^𝜖).

We recall that by [17, Theorem 2.1] and [17, Theorem 2.3] (3) has a unique solution 𝑢^𝜖∈𝑊^1,𝑝(Ω^𝜖).

In the analysis of the limiting behavior of 𝑢^𝜖, it will be useful to introduce the domain Ω∶=𝜔×(0,1) which is independent of 𝜖 and is obtained from Ω^𝜖 by the change of coordinates 𝒯^𝜖∶Ω⟶Ω^𝜖𝑥,𝑦↦𝑥,𝜖𝑔(𝑥)𝑦. (4)

Such change of coordinates induces an isomorphism from 𝑊^𝑚,𝑝(Ω^𝜖) onto 𝑊^𝑚,𝑝(Ω) by 𝑢↦^Φ𝜖𝑣∶=𝑢∘𝒯^𝜖, (5) with partial derivatives related by 𝑢_𝑥𝑖=𝑣_𝑥𝑖−𝑦𝑔_𝑥𝑖𝑔𝑣_𝑦,𝑖=1,…,𝑛,𝑢_𝑦=1𝑣𝜖𝑔^𝑦. (6)

In this new system of coordinates, (2) is written as −1𝑔divℒ_𝜖𝑣^𝑝−2𝐵_𝜖𝑣+|𝑣|^𝑝−2𝑣=ℎ^𝜖,in𝐵Ω,_𝜖𝑣⋅𝜂=0,on𝜕Ω, (7) where ℎ^𝜖∶=Φ^𝜖(𝑓^𝜖), ℒ_𝜖𝑣=(𝑣_𝑥1−(𝑔_𝑥1𝑦𝑣_𝑦/𝑔),…,𝑣_𝑥𝑛−(𝑔_𝑥𝑛𝑦𝑣_𝑦/𝑔),𝑣_𝑦/𝜖𝑔), 𝐵_𝜖𝑣=𝑔𝑣_𝑥1−𝑦𝑔_𝑥1𝑣_𝑦⋮𝑔𝑣_𝑥𝑛−𝑦𝑔_𝑥𝑛𝑣_𝑦−∑^𝑛_𝑖=1𝑦𝑔_𝑥𝑖𝑣_𝑥𝑖+1𝜖²𝑔1+∑^𝑛_𝑖=1𝜖𝑦𝑔_𝑥𝑖²𝑣_𝑦, (8) and 𝜂 denotes the unit outward normal vector field to 𝜕Ω.

Noticing that 𝑢∈𝑊^1,𝑝(Ω^𝜖) is a solution of (2) if and only if 𝑣∶=Φ(𝑢)∈𝑊^1,𝑝(Ω) is a solution of (7), the rest of this paper is dedicated to the study of the limiting behavior of the solutions of (7); that is, functions 𝑣∈𝑊^1,𝑝(Ω) such that _Ωℒ_𝜖𝑣^𝑝−2𝐵_𝜖𝑣⋅∇𝜑+𝑔|𝑣|^𝑝−2𝑣𝜑𝑑𝑥𝑑𝑦=_Ω𝑔ℎ^𝜖𝜑𝑑𝑥𝑑𝑦, (9) for all 𝜑∈𝑊^1,𝑝(Ω).

Due to the nature of this specific domain perturbation, solutions of (7) tend not to depend “so much” on the variable 𝑦 as 𝜖≈0. This suggests comparing such solutions with solutions of the following equation: −1𝑔div𝑔|∇𝑣|^𝑝−2∇𝑣+|𝑣|^𝑝−2𝑣=ℎ,in𝜔,𝜕𝑣𝜕𝜈=0,on𝜕𝜔, (10) for some appropriate ℎ∈𝐿^𝑞(𝜔), where 𝜈 denotes the unit outward normal vector field to 𝜕𝜔.

Again by [17, Theorem 2.1] and [17, Theorem 2.3] we can derive a unique solution 𝑣⁰∈𝑊^1,𝑝(𝜔) of (10).

The paper is organized as follows. In Section 2 we set up the appropriate functional framework which will be used to compare the problems (7) and (10), and in the subsequent Section 3, we formulate and prove the convergence results.

2. Preliminaries

Stressing for the fact that the domains Ω^𝜖 vary in accordance with the small parameter 𝜖, collapsing themselves to a lower dimensional subset as 𝜖 goes to 0, we perform a dilatation on the Lebesgue measure in ℝ^𝑛+1 in order to preserve the relative capacity of measurable subsets of Ω^𝜖. Thus, we consider the Lebesgue space, 𝐿^𝑝(Ω^𝜖), endowed with the equivalent norm |||𝑢|||_𝐿^𝑝_(Ω^𝜖₎=𝜖⁻¹_Ω𝜖|𝑢|^𝑝𝑑𝑥𝑑𝑦^1/𝑝 (11) and the Sobolev space, 𝑊^1,𝑝(Ω^𝜖), endowed with the equivalent norm |||𝑢|||_𝑊^1,𝑝_(Ω^𝜖₎=𝜖⁻¹_Ω𝜖∇_𝑥𝑢^𝑝+∇_𝑦𝑢^𝑝+|𝑢|^𝑝𝑑𝑥𝑑𝑦^1/𝑝. (12)

We also consider equivalent norms in 𝐿^𝑝(Ω) and in 𝑊^1,𝑝(Ω) given, respectively, by |||𝑢|||_𝐿^𝑝_(Ω)=_Ω𝑔(𝑥)|𝑢|^𝑝𝑑𝑥𝑑𝑦^1/𝑝,|||𝑢|||_𝜖∶=_Ω𝑔(𝑥)∇_𝑥𝑢^𝑝+1𝜖^𝑝∇_𝑦𝑢^𝑝+|𝑢|^𝑝𝑑𝑥𝑑𝑦^1/𝑝. (13)

It is immediate from these definitions that |||𝑢|||_𝐿^𝑝_(Ω^𝜖₎=Φ^𝜖(𝑢)_𝐿𝑝(Ω), (14) and there exist positive constants 𝑐₁, 𝑐₂ such that 𝑐₁|||𝑢|||_𝑊^1,𝑝_(Ω^𝜖₎≤Φ^𝜖(𝑢)_𝜖≤𝑐₂|||𝑢|||_𝑊^1,𝑝_(Ω^𝜖₎. (15)

Finally, since we need to compare functions defined in different domains, for example, Ω and 𝜔, is natural to introduce the following operators.

Average projector:𝑀∶𝑊^1,𝑝(Ω)⟶𝑊^1,𝑝(𝜔),𝑀(𝑢)(𝑥)=¹₀𝑢𝑥,𝑦𝑑𝑦. (16)

Extension operator:𝐸∶𝑊^1,𝑝(𝜔)⟶𝑊^1,𝑝(Ω),𝐸(𝑢)𝑥,𝑦=𝑢(𝑥). (17)

3. Convergence

Given 𝑓^𝜖∈𝐿^𝑞(Ω^𝜖), 1<𝑞<2, if 𝑢^𝜖∈𝑊^1,𝑝(Ω^𝜖) is a solution of (2) we have by Hölder's inequality that 𝑢^𝜖𝑝_{𝑊1,𝑝(Ω𝜖)}=_Ω𝜖𝑓^𝜖𝑢^𝜖≤𝑓𝑑𝑥𝑑𝑦_{𝜖𝐿𝑞(Ω𝜖)}𝑢^𝜖_{𝐿𝑝(Ω𝜖)}≤𝑓^𝜖_{𝐿𝑞(Ω𝜖)}𝑢^𝜖_{𝑊1,𝑝(Ω𝜖)}. (18) This shows that 𝑢^𝜖_𝑊^𝑝−1_{1,𝑝(Ω𝜖)}≤𝑓^𝜖_{𝐿𝑞(Ω𝜖)}, (19) which gives us the following a priori estimate for solutions of (7): 𝑣^𝜖_𝜖^𝑝−1≤𝑐₂^𝑝−1ℎ^𝜖_𝐿𝑞(Ω), (20) where 𝑣^𝜖=Φ^𝜖(𝑢^𝜖) and ℎ^𝜖=Φ^𝜖(𝑓^𝜖).

Such a priori estimate is the essence of the following lemma.

Lemma 2. Let ℎ^𝜖∈𝐿^𝑞(Ω), 1<𝑞<2, such that ℎ^𝜖⇀ℎ⁰ in 𝐿^𝑞(Ω). If 𝑣^𝜖∈𝑊^1,𝑝(Ω) is a solution of (7), there exists 𝑣⁰∈𝑊^1,𝑝(𝜔) solution of (10) with 𝑓=𝑀(ℎ⁰), such that up to subsequence 𝑣⟶^𝜖𝜖→0𝐸𝑣⁰,𝑤𝑒𝑎𝑘𝑙𝑦-𝑊^1,𝑝(Ω)𝑎𝑛𝑑𝑠𝑡𝑟𝑜𝑛𝑔𝑙𝑦-𝐿^𝑝(Ω). (21)

Proof. It follows from (20) that ‖𝑣^𝜖‖_𝑊^𝑝−1_1,𝑝(Ω)⩽𝑐, for some constant 𝑐 independent of 𝜖. Since 𝑊^1,𝑝(Ω) is a reflexive space and 𝐿^𝑝(Ω)↪𝑊^1,𝑝(Ω) compactly, taking subsequence if necessary, there exists 𝑣⁰̃∈𝑊^1,𝑝(Ω) such that 𝑣−−−−→^𝜖𝜖→0𝑣⁰̃, weakly in 𝑊^1,𝑝(Ω) and strongly in 𝐿^𝑝(Ω). Moreover, noticing that {|||𝑣^𝜖|||_𝜖}_𝜖 is bounded, one has that ‖∇_𝑦𝑣^𝜖‖_𝐿𝑝(Ω)=𝑂(𝜖). Therefore ∇_𝑦𝑣−−−−→^𝜖𝜖→00 in 𝐿^𝑝(Ω), which means that ∇_𝑦𝑣⁰̃=0 almost everywhere in Ω. This implies the existence of 𝑣⁰∈𝑊^1,𝑝(𝜔), such that 𝑣⁰̃(𝑥,𝑦)=𝑣⁰(𝑥), almost everywhere in Ω.
On the other hand, noticing that 𝐸(𝜑)∈𝑊^1,𝑝(Ω) whenever 𝜑∈𝑊^1,𝑝(𝜔) and since ∇_𝑦𝐸(𝜑)=0, it follows from the weak convergence |𝑣^𝜖|^𝑝−2𝑣^𝜖⇀|𝑣⁰|^𝑝−2𝑣⁰ in 𝐿^𝑞(Ω) that 𝑔𝑀ℎ_𝜔⁰𝜑𝑑𝑥⟵^𝜖→0_Ω𝑔ℎ^𝜖𝐸𝜑𝑑𝑥𝑑𝑦=_Ωℒ_𝜖𝑣^𝜖𝑝−2𝐵_𝜖𝑣^𝜖⋅∇𝐸𝜑+𝑔𝑣^𝜖𝑝−2𝑣^𝜖𝐸𝜑𝑑𝑥𝑑𝑦_𝜖→0⟶_Ω𝑔∇𝑣̃^0𝑝−2∇𝑣⁰̃𝜑𝑣̃⋅∇𝐸+𝑔^0𝑝−2𝑣⁰̃𝐸𝜑=𝑑𝑥𝑑𝑦_𝜔𝑔∇𝑣^0𝑝−2∇𝑣⁰⋅∇𝜑+𝑔𝑣^0𝑝−2𝑣⁰𝜑𝑑𝑥, (22) for all 𝜑∈𝑊^1,𝑝(𝜔).

As a consequence of the following theorem and inspired by [8, 15], we obtain the convergence of the family of solutions 𝑣^𝜖 in the norm |||⋅|||_𝜖.

Theorem 3. Let ℎ^𝜖, 𝑣^𝜖, ℎ⁰, and 𝑣⁰ be as in Lemma 2. Then lim_𝜖→0𝑣^𝜖−𝐸𝑣⁰_{𝑊1,𝑝(Ω)}=0. (23)

Proof. It follows from the weak convergence 𝑣^𝜖⇀𝑣⁰̃ in 𝑊^1,𝑝(Ω) (obtained in Lemma 2) that _Ω𝑔∇_𝑥𝑣⁰̃|^𝑝+𝑣⁰̃|^𝑝𝑑𝑥𝑑𝑦⩽liminf_𝜖→0Ω𝑔∇_𝑥𝑣^𝜖𝑝+∇_𝑦𝑣^𝜖𝑝+𝑣^𝜖𝑝𝑑𝑥𝑑𝑦⩽limsup_𝜖→0Ω𝑔∇_𝑥𝑣^𝜖𝑝+∇_𝑦𝑣^𝜖𝑝+𝑣^𝜖𝑝𝑑𝑥𝑑𝑦⩽lim_𝜖→0Ω𝑔∇_𝑥𝑣^𝜖𝑝+1𝜖^𝑝∇_𝑦𝑣^𝜖𝑝+𝑣^𝜖𝑝𝑑𝑥𝑑𝑦=lim_𝜖→0Ω𝑔ℎ^𝜖𝑣^𝜖𝑑𝑥𝑑𝑦=_Ω𝑔ℎ⁰𝑣⁰̃=_Ω𝑔∇𝑣̃^0𝑝+𝑣̃^0𝑝𝑑𝑥𝑑𝑦. (24) Recalling that for 𝑣⁰̃, [∫_Ω𝑔(|∇_𝑥𝑣⁰̃|^𝑝+|𝑣⁰̃|^𝑝)𝑑𝑥𝑑𝑦]^1/𝑝 is an equivalent norm in 𝑊^1,𝑝(Ω), this proves the statement.

Corollary 4. Let 𝑣^𝜖 and 𝑣⁰ be as in Theorem 3. Then lim_𝜖→0𝑣^𝜖−𝐸𝑣⁰_𝜖=0. (25)

Proof. According to Theorem 3_Ω𝑔∇_𝑥𝑣̃^0𝑝+𝑣̃^0𝑝𝑑𝑥𝑑𝑦=lim_𝜖→0Ω𝑔∇_𝑥𝑣^𝜖𝑝+1𝜖^𝑝∇_𝑦𝑣^𝜖𝑝+𝑣^𝜖𝑝𝑑𝑥𝑑𝑦, (26) which implies that lim_𝜖→0(1/𝜖^𝑝)|∇_𝑦𝑣^𝜖|^𝑝=0.

Remark 5. We would like to recall that Hale and Raugel in [12] obtained in the case 𝑝=2 as the limiting problem for the similar equation −1𝑔divℬ_𝜖𝑣+𝑣=ℎ^𝜖,in𝐵Ω,_𝜖𝑣⋅𝜂=0,on𝜕Ω, (27) the problem −1𝑔div𝑔∇𝑣+𝑣=ℎ,in𝜔,𝜕𝑣𝜕𝜈=0,on𝜕𝜔. (28)
After the previous considerations we point out the robustness of the structure of this limiting problem in the following sense: considering 𝑝 as a parameter as well as 𝜖, allowing 𝑝→2 and 𝜖→0, independent of the order of the convergence, we obtain the same limiting problem, namely, (28). We summarize that in the following commutative diagram: Equation(7)−−−−→^𝑝→2Equation()↺27𝜖→0↓↓𝜖→0Equation(10)−−−−→_𝑝→2Equation(28) (29)

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author thanks Professor Olímpio Myagaki for stimulating conversations and useful remarks. The author also thanks the referees for their suggestions which led to an improvement of the text. This work was partially supported by FAPESP no. 2012/06753-8, FUNDUNESP no. 0135812, and FUNDUNESP-PROPe no. 0019/008/13, Brazil.