# Behavior of thep-Laplacian on Thin Domains

Authors: Ricardo P. Silva
Journal: International Journal of Differential Equations
 (2013)

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## Abstract

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


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## Body

## 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, letω be a smooth bounded domain in ℝn, 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 Ωϵ⊂ℝn+1 defined by
(1)Ωϵ:={(x,y)∈ℝn×ℝ:x∈ω,0<y<ϵg(x)}.
The aim of this paper is to characterize the limiting problem (ϵ=0) for the family of elliptic equations
(2)-Δpu+|u|p-2u=fϵ,inΩϵ,∂u∂ηϵ=0,on∂Ωϵ,
where p>2, fϵ∈Lq(Ωϵ), (1/p)+(1/q)=1, Δpu:=div(|∇u|p-2∇u) denotes the p-Laplacian operator and ηϵ denotes the outward unitary normal vector field to ∂Ωϵ.Definition 1.
Givenfϵ∈Lq(Ωϵ), 1<q<2, one says that u∈W1,p(Ωϵ), (1/p)+(1/q)=1 is a solution of (2) if
(3)∫Ωϵ‍(|∇u|p-2∇u·∇φ+|u|p-2uφ)dxdy=∫Ωϵ‍fϵφdxdy,
for all φ∈W1,p(Ωϵ).We recall that by [17, Theorem 2.1] and [17, Theorem 2.3] (3) has a unique solution uϵ∈W1,p(Ωϵ).In the analysis of the limiting behavior ofuϵ, it will be useful to introduce the domain Ω:=ω×(0,1) which is independent of ϵ and is obtained from Ωϵ by the change of coordinates
(4)𝒯ϵ:Ω⟶Ωϵ(x,y)↦(x,ϵg(x)y).Such change of coordinates induces an isomorphism fromWm,p(Ωϵ) onto Wm,p(Ω) by
(5)u↦Φϵv:=u∘𝒯ϵ,
with partial derivatives related by
(6)uxi=vxi-ygxigvy,i=1,…,n,uy=1ϵgvy.In this new system of coordinates, (2) is written as (7)-1gdiv(|ℒϵv|p-2Bϵv)+|v|p-2v=hϵ,inΩ,Bϵv·η=0,on∂Ω,
where hϵ:=Φϵ(fϵ), ℒϵv=(vx1-(gx1yvy/g),…,vxn-(gxnyvy/g),vy/ϵg),
(8)Bϵv=[gvx1-ygx1vy⋮gvxn-ygxnvy-∑i=1n‍ygxivxi+1ϵ2g(1+∑i=1n‍(ϵygxi)2)vy],
and η denotes the unit outward normal vector field to ∂Ω.Noticing thatu∈W1,p(Ωϵ) is a solution of (2) if and only if v:=Φ(u)∈W1,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∈W1,p(Ω) such that
(9)∫Ω‍(|ℒϵv|p-2Bϵv·∇φ+g|v|p-2vφ)dxdy=∫Ω‍ghϵφdxdy,
for all φ∈W1,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:
(10)-1gdiv(g|∇v|p-2∇v)+|v|p-2v=h^,inω,∂v∂ν=0,on∂ω,
for some appropriate h^∈Lq(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∈W1,p(ω) of (10).The paper is organized as follows. In Section2 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 domainsΩϵ 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 ℝn+1 in order to preserve the relative capacity of measurable subsets of Ωϵ. Thus, we consider the Lebesgue space, Lp(Ωϵ), endowed with the equivalent norm
(11)|||u|||Lp(Ωϵ)=[ϵ-1∫Ωϵ‍|u|pdxdy]1/p
and the Sobolev space, W1,p(Ωϵ), endowed with the equivalent norm
(12)|||u|||W1,p(Ωϵ)=[ϵ-1∫Ωϵ‍(|∇xu|p+|∇yu|p+|u|p)dxdy]1/p.We also consider equivalent norms inLp(Ω) and in W1,p(Ω) given, respectively, by
(13)|||u|||Lp(Ω)=[∫Ω‍g(x)|u|pdxdy]1/p,|||u|||ϵ:=[∫Ω‍g(x)(|∇xu|p+1ϵp|∇yu|p+|u|p)dxdy]1/p.It is immediate from these definitions that(14)|||u|||Lp(Ωϵ)=|||Φϵ(u)|||Lp(Ω),
and there exist positive constants c1, c2 such that
(15)c1|||u|||W1,p(Ωϵ)≤|||Φϵ(u)|||ϵ≤c2|||u|||W1,p(Ωϵ).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:(16)M:W1,p(Ω)⟶W1,p(ω),M(u)(x)=∫01‍u(x,y)dy.Extension operator:(17)E:W1,p(ω)⟶W1,p(Ω),E(u)(x,y)=u(x).


## 3. Convergence

Givenfϵ∈Lq(Ωϵ), 1<q<2, if uϵ∈W1,p(Ωϵ) is a solution of (2) we have by Hölder's inequality that
(18)∥uϵ∥W1,p(Ωϵ)p=∫Ωϵfϵuϵdxdy≤∥fϵ∥Lq(Ωϵ)∥uϵ∥Lp(Ωϵ)≤∥fϵ∥Lq(Ωϵ)∥uϵ∥W1,p(Ωϵ).
This shows that
(19)|||uϵ|||W1,p(Ωϵ)p-1≤|||fϵ|||Lq(Ωϵ),
which gives us the following a priori estimate for solutions of (7):
(20)|||vϵ|||ϵp-1≤c2p-1|||hϵ|||Lq(Ω),
where vϵ=Φϵ(uϵ) and hϵ=Φϵ(fϵ).Such a priori estimate is the essence of the following lemma.Lemma 2.
Lethϵ∈Lq(Ω), 1<q<2, such that hϵ⇀h0 in Lq(Ω). If vϵ∈W1,p(Ω) is a solution of (7), there exists v0∈W1,p(ω) solution of (10) with f^=M(h0), such that up to subsequence (21)vϵ⟶ϵ→0Ev0,weakly-W1,p(Ω)andstrongly-Lp(Ω).Proof.
It follows from (20) that ∥vϵ∥W1,p(Ω)p-1⩽c, for some constant c independent of ϵ. Since W1,p(Ω) is a reflexive space and Lp(Ω)↪W1,p(Ω) compactly, taking subsequence if necessary, there exists v~0∈W1,p(Ω) such that vϵ→ϵ→0v~0, weakly in W1,p(Ω) and strongly in Lp(Ω). Moreover, noticing that {|||vϵ|||ϵ}ϵ is bounded, one has that ∥∇yvϵ∥Lp(Ω)=O(ϵ). Therefore ∇yvϵ→ϵ→00 in Lp(Ω), which means that ∇yv~0=0 almost everywhere in Ω. This implies the existence of v0∈W1,p(ω), such that v~0(x,y)=v0(x), almost everywhere in Ω.
On the other hand, noticing thatE(φ)∈W1,p(Ω) whenever φ∈W1,p(ω) and since ∇yE(φ)=0, it follows from the weak convergence |vϵ|p-2vϵ⇀|v0|p-2v0 in Lq(Ω) that
(22)∫ω‍gM(h0)φdx⟵ϵ→0∫Ω‍ghϵE(φ)dxdy=∫Ω‍(|ℒϵvϵ|p-2Bϵvϵ·∇E(φ)+g|vϵ|p-2vϵE(φ))dxdy⟶ϵ→0∫Ω‍(E(φ)g|∇v~0|p-2∇v~0·∇E(φ)+g|v~0|p-2v~0E(φ))dxdy=∫ω‍(g|∇v0|p-2∇v0·∇φ+g|v0|p-2v0φ)dx,
for all φ∈W1,p(ω).As a consequence of the following theorem and inspired by [8, 15], we obtain the convergence of the family of solutions vϵ in the norm |||·|||ϵ.Theorem 3.
Lethϵ, vϵ, h0, and v0 be as in Lemma 2. Then
(23)limϵ→0∥vϵ-Ev0∥W1,p(Ω)=0.Proof.
It follows from the weak convergencevϵ⇀v~0 in W1,p(Ω) (obtained in Lemma 2) that
(24)∫Ω‍g(|∇xv~0|p+|v~0|p)dxdy⩽liminfϵ→0∫Ω‍g(|∇xvϵ|p+|∇yvϵ|p+|vϵ|p)dxdy⩽limsupϵ→0∫Ω‍g(|∇xvϵ|p+|∇yvϵ|p+|vϵ|p)dxdy⩽limϵ→0∫Ω‍g(|∇xvϵ|p+1ϵp|∇yvϵ|p+|vϵ|p)dxdy=limϵ→0∫Ω‍ghϵvϵdxdy=∫Ω‍gh0v~0=∫Ω‍g(|∇v~0|p+|v~0|p)dxdy.
Recalling that for v~0, [∫Ω‍g(|∇xv~0|p+|v~0|p)dxdy]1/p is an equivalent norm in W1,p(Ω), this proves the statement.Corollary 4.
Letvϵ and v0 be as in Theorem 3. Then
(25)limϵ→0|||vϵ-Ev0|||ϵ=0.Proof.
According to Theorem3(26)∫Ω‍g(|∇xv~0|p+|v~0|p)dxdy=limϵ→0∫Ω‍g(|∇xvϵ|p+1ϵp|∇yvϵ|p+|vϵ|p)dxdy,
which implies that limϵ→0(1/ϵp)|∇yvϵ|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
(27)-1gdiv(ℬϵv)+v=hϵ,inΩ,Bϵv·η=0,on∂Ω,
the problem
(28)-1gdiv(g∇v)+v=h^,inω,∂v∂ν=0,on∂ω.
After the previous considerations we point out the robustness of the structure of this limiting problem in the following sense: consideringp 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:
(29)Equation(7)→p→2Equation(27)ϵ→0↓↺↓ϵ→0Equation(10)→p→2p→2Equation(28)



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*Source: 210270-2013-11-18.xml*