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The p-Laplacian in thin channels with locally periodic rough boundaries

J. C. Nakasato, M. C. Pereira

Abstract

In this work we analyze the asymptotic behavior of the solutions of the $p$-Laplacian equation with homogeneous Neumann boundary conditions set in bounded thin domains as $$R^\varepsilon=\left\lbrace(x,y)\in\mathbb{R}^2:x\in(0,1)\mbox{ and }0<y<\varepsilon G\left(x,{x}/{\varepsilon}\right)\right\rbrace.$$ We take a smooth function $G:(0,1)\times\mathbb{R} \mapsto \mathbb{R}$, $L$-periodic in the second variable, which allows us to consider locally periodic oscillations at the upper boundary. The thin domain situation is established passing to the limit in the solutions as the positive parameter $\varepsilon$ goes to zero.

The p-Laplacian in thin channels with locally periodic rough boundaries

Abstract

In this work we analyze the asymptotic behavior of the solutions of the -Laplacian equation with homogeneous Neumann boundary conditions set in bounded thin domains as We take a smooth function , -periodic in the second variable, which allows us to consider locally periodic oscillations at the upper boundary. The thin domain situation is established passing to the limit in the solutions as the positive parameter goes to zero.

Paper Structure

This paper contains 9 sections, 15 theorems, 181 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unfolding operator
  4. Purely periodic
  5. Locally Periodic Unfolding
  6. A domain dependence result
  7. The piecewise periodic case
  8. The locally periodic case
  9. Appendix

Key Result

Theorem 1.1

Let $u_{\varepsilon}$ be the solution of problem with $f^\varepsilon\in L^{p'}(R^\varepsilon)$ uniformly bounded. Suppose that satisfies $\hat{f}^\varepsilon\rightharpoonup\hat{f}\hbox{weakly in}L^{p'}(0,1).$ Then, there exists $u\in W^{1,p}(0,1)$ such that with $u$ satisfying the homogenized equation where the homogenized coefficients $q(x)$ and $r(x)$ are given by coeq.

Figures (2)

  • Figure 1: A locally periodic thin channel with rough boundary.
  • Figure 2: A piecewise periodic thin domain.

Theorems & Definitions (31)

  • Theorem 1.1
  • Proposition 2.1
  • Corollary 2.1.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • Remark 2.1
  • Proposition 2.6
  • ...and 21 more