Boundary and interior layer behavior of solutions of some singularly perturbed semilinear elliptic boundary value problems
type: Item
instance of: scholarly article
title: Boundary and interior layer behavior of solutions of some singularly perturbed semilinear elliptic boundary value problems
zbMATH Open document ID: 0549.35046
author: Albert James DeSanti
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1985-01-01T00:00:00Z
review text: We study the asymptotic behavior (as \(\epsilon\to 0)\) of solutions of the singularly perturbed semilinear elliptic boundary value problem \((*)\quad\epsilon^ 2\Delta u=h(\underset \tilde{} x,u)\) in D, \(u(\underset \tilde{} x,\epsilon)=f(\underset \tilde{} x)\) on B, where \(\Delta\) is the Laplacian operator, D is an open set in \(R^ n\) bounded by the smooth manifold B, and h and f are prescribed smooth functions. Using a method based on the theory of partial differential inequalities we show that under explicit conditions on h and f the problem (*) has a solution which exhibits boundary layer behavior and one of three distinct types of interior layer behavior. The conditions on h and f are given in terms of a potential energy functional \(V(\underset \tilde{} x,u)=\int^{u}_{0}h(\underset \tilde{} x,s)ds.\) Our method of proof also allows the construction of approximate solutions of problem (*) which serve as lower and upper bounds for the exact solutions.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35A35
Mathematics Subject Classification ID: 35R45
zbMATH DE Number: 3875825
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: singularly perturbed semilinear elliptic boundary value problem
zbMATH Keywords: partial differential inequalities
zbMATH Keywords: boundary layer
zbMATH Keywords: approximate solutions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3875825