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Homogenization of a nonlinear monotone problem in a locally periodic domain via unfolding method

S. Aiyappan, G. Cardone, C. Perugia, R. Prakash

Abstract

In this paper, the asymptotic behavior of the solutions of a monotone problem posed in a locally periodic oscillating domain is studied. Nonlinear monotone boundary conditions are imposed on the oscillating part of the boundary whereas the Dirichlet condition is considered on the smooth separate part. Using the unfolding method, under natural hypothesis on the regularity of the domain, we prove the weak $L^2$-convergence of the zero-extended solutions of the nonlinear problem and their flows to the solutions of a limit distributional problem.

Homogenization of a nonlinear monotone problem in a locally periodic domain via unfolding method

Abstract

In this paper, the asymptotic behavior of the solutions of a monotone problem posed in a locally periodic oscillating domain is studied. Nonlinear monotone boundary conditions are imposed on the oscillating part of the boundary whereas the Dirichlet condition is considered on the smooth separate part. Using the unfolding method, under natural hypothesis on the regularity of the domain, we prove the weak -convergence of the zero-extended solutions of the nonlinear problem and their flows to the solutions of a limit distributional problem.

Paper Structure

This paper contains 6 sections, 5 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting of the problem and main result
  3. Preliminaries
  4. The periodic unfolding operator
  5. Compactness results
  6. Proof of Theorem \ref{['tm:homogenization']}.

Key Result

Theorem 2.1

Under the assumptions $H1)\div H3)$, let $u_\varepsilon \in H^1(\Omega^\varepsilon, \Gamma_b)$ be the sequence of solutions to NHP-1. Then, there exist $u_0\in W(\Omega, \Gamma_b)$ and $\overline q \in L^2(\Omega^+)$ such that, as $\varepsilon$ tends to zero, the following convergences hold where $\sim$ denotes the classical extension to zero and the pair $(u_0, \overline q)\in W(\Omega, \Gamma_b

Figures (1)

  • Figure 1: A locally periodic domain $\Omega^\varepsilon$ (a), $\varepsilon = 1/8$, and the corresponding homogeneous domain $\Omega$ (b), with $\Gamma_-$ marked with a dashed line separating the regions $\Omega_+$ and $\Omega_-$.

Theorems & Definitions (11)

  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Definition 3.1
  • Lemma 3.1
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 1 more