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The asymptotic behavior for divergence elliptic equations in exterior domains with periodic coefficients

Lichun Liang

Abstract

In this paper, we investigate the asymptotic behavior of solutions for divergence linear elliptic equations in exterior domains with periodic coefficients. Consequently, we generalise the Liouville type result firstly established by Avellaneda and Lin.

The asymptotic behavior for divergence elliptic equations in exterior domains with periodic coefficients

Abstract

In this paper, we investigate the asymptotic behavior of solutions for divergence linear elliptic equations in exterior domains with periodic coefficients. Consequently, we generalise the Liouville type result firstly established by Avellaneda and Lin.
Paper Structure (3 sections, 3 theorems, 45 equations)

This paper contains 3 sections, 3 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. The proof of Theorem \ref{['th1']}
  3. The proof of Theorem \ref{['th2']}

Key Result

Theorem 1

Assume that the coefficients $a_{ij}$ satisfy (1)-(3) and are Lipschitz continuity. Let $u\in W_{loc}^{1,2}(\mathbb{R}^n)$ be a solution of with for all $r \geq 1$ and some integer $N\geq 0$ and constant $C>0$. Then where $p_\nu(x)$ are periodic and Hölder continuous. Moreover, when $|\nu|=N$, the coefficients $p_\nu(x)$ are constants.

Theorems & Definitions (7)

  • Theorem : Avellaneda-Lin
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • proof : Proof of Theorem \ref{['th1']}
  • proof : Proof of Theorem \ref{['th2']}