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Poisson equation in domains with concentrated holes

Hiroto Ishida

Abstract

We consider solutions $u^\varepsilon$ of Poisson problems with the Dirichlet condition on domains $Ω_\varepsilon$ with holes concentrated at subsets of a domain $Ω$ non-periodically. We show $u^\varepsilon$ converges to a solution of a Poisson problem with a simple function potential. This is a generalized result of a sample model given by Cioranescu and Murat (1997). They showed a result for case that holes are distributed at $Ω$ periodically.

Poisson equation in domains with concentrated holes

Abstract

We consider solutions of Poisson problems with the Dirichlet condition on domains with holes concentrated at subsets of a domain non-periodically. We show converges to a solution of a Poisson problem with a simple function potential. This is a generalized result of a sample model given by Cioranescu and Murat (1997). They showed a result for case that holes are distributed at periodically.
Paper Structure (10 sections, 12 theorems, 33 equations, 2 figures)

This paper contains 10 sections, 12 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Known results
  3. Assumption and the main result
  4. Assumption
  5. Result
  6. Outline of proof
  7. Proof
  8. Approximation of sets by tiles
  9. Error corrector
  10. Proof of

Key Result

Theorem 1

Under the assumptions as in ass, $u^\varepsilon$ in solus converges to $u$ weakly in $H_0^1(\Omega)$ and the limit $u$ solves PDE with

Figures (2)

  • Figure 1: A domain $\Omega$ and holes $T_\varepsilon.$
  • Figure 2: Construction of holes $T_\varepsilon$ with $m=2,~N_1=6,~N_2=2.$

Theorems & Definitions (27)

  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 1
  • Remark 2
  • Theorem 2: CM
  • Lemma 1
  • proof
  • Definition 3
  • Lemma 2
  • ...and 17 more