A singularly perturbed linear eigenvalue problem in \(C^1\) domains.
type: Item
instance of: scholarly article
title: A singularly perturbed linear eigenvalue problem in \(C^1\) domains.
zbMATH Open document ID: 1061.35061
author: Yuan Lou
author: Meijun Zhu
published in: Pacific Journal of Mathematics
publication date: 2004-09-17T00:00:00Z
review text: For any \(\gamma>0\), set \[ \Lambda(\gamma)=\sup_{u\in H^1(\Omega)\setminus\{0\}}\frac{\gamma\int_{\partial\Omega}u^2-\int_\Omega| \nabla u| ^2}{\int_\Omega u^2}, \] where \(\Omega\) is a bounded domain in \(\mathbb R^n\) with boundary \(\partial\Omega\in C^1\). The supremum is attained by some positive function \(u_\gamma\in H^1(\Omega)\) , which is a weak solution of \[ \Delta u=\Lambda(\gamma)u\quad \text{in } \Omega,\qquad \frac{\partial u}{\partial\nu}=\gamma u\quad \text{on }\partial\Omega, \] where \(\nu\) is the outward unit normal vector on \(\partial\Omega\). The goal of this paper is to understand the asymptotic behavior of \(\Lambda(\gamma)\) as \(\gamma\to\infty\). Since \(\Lambda(\gamma)\to\infty\) when \(\gamma\to\infty\), this problem can be viewed as a singularly perturbed linear eigenvalue problem. The following theorems are proved. Theorem 1. \[ \lim_{\gamma\to\infty}\frac{\Lambda(\gamma)}{\gamma^2}=1 \] holds for any bounded \(C^1\) domain. Theorem 2. If \(a>1\), then \[ \Delta u=au \quad \text{in } \mathbb R_+^n,\qquad \frac{\partial u}{\partial x_n}=-u\quad \text{on }\partial \mathbb R_+^n, \] has no bounded nontrivial solution. Here \(a\) is the limit of \(\frac{\Lambda(\gamma)}{\gamma^2}\) (subject to a subsequence) as \(\gamma\to\infty\).
reviewed by: Andrey Ivanovic Sedov
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 2101156
zbMATH Keywords: bounded domain
zbMATH Keywords: weak solution
zbMATH Keywords: asymptotic behavior
MaRDI profile type: Publication
full work available at URL: pjm.2004.214.323
OpenAlex ID: W2006810711
DOI: 10.2140/PJM.2004.214.323
description: scientific article; zbMATH DE number 2101156