The Dirichlet problem for an elliptic operator in a domain with cusp point
type: Item
instance of: scholarly article
title: The Dirichlet problem for an elliptic operator in a domain with cusp point
zbMATH Open document ID: 0939.35060
author: Jean-Luc Steux
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2000-06-26T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73411
review text: The paper is devoted to the study of qualitative properties of solutions to the homogeneous Dirichlet boundary value problem for a properly elliptic operator of the order \(2m\) with smooth coefficients on a plane domain \(U\) with a boundary having a cusp point in the origin \(0\). The boundary of \(U\) in a sufficiently small neighbourhood of \(0\) is described as a graph of two smooth functions \(\varphi_1, \varphi_2\) such that \[ \varphi_1, \varphi_2 \in C^1([0,\epsilon]) \cap C^{\infty}(]0,\epsilon]), \] \[ \varphi_1 < \varphi_2,\quad \varphi_1(0) = \varphi_2(0) = 0,\quad \varphi_1'(0) = \varphi_2'(0) = 0. \] The author gives conditions on \(\varphi_1, \varphi_2\) guaranteeing the smoothness of solutions or eigenfunctions of the problem and, if a solution is not smooth, he gives an asymptotic expansion near the cusp point.
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 1066075
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: properly elliptic operator
zbMATH Keywords: plane domain with a cusp point
zbMATH Keywords: asymptotics of a solution
zbMATH Keywords: regularity of solutions
reviewed by: J. Stará
MaRDI profile type: Publication
OpenAlex ID: W2315242398
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DOI: 10.5802/AFST.859
description: scientific article; zbMATH DE number 1066075