Normal solvability of the Dirichlet problem for the vibrating string equation in domains with nonsmooth boundaries
type: Item
instance of: scholarly article
title: Normal solvability of the Dirichlet problem for the vibrating string equation in domains with nonsmooth boundaries
zbMATH Open document ID: 0806.35102
DOI: 10.1007/BF00970894
published in: Siberian Mathematical Journal
publication date: 1993-03-18T00:00:00Z
review text: The article is devoted to the Dirichlet problems in a bounded convex domain \(\Omega \subset \mathbb{R}^ 2\) with piecewise smooth boundary for the equations \(\overline {\mathcal L}_ \lambda u = f\) and \(\overline {\mathcal L}_ \lambda^* v = g\) \((f,g \in L_ 2)\), where \({\mathcal L}_ \lambda u = (1 + \lambda) u_{xx} - (1 - \lambda) u_{yy}\) \((\lambda \in (- 1,1))\). The main goal is to study the winding number of some auxiliary homeomorphism \(F_ \lambda\) that is defined with \(\Omega\); in particular necessary and sufficient conditions are given under which the winding number is rational for all \(\lambda\) or for all, except a finite number of \(\lambda\)'s. As a corollary some results on normal solvability for the problems above are presented.
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 130810
zbMATH Keywords: nonsmooth boundaries
zbMATH Keywords: Dirichlet problems
zbMATH Keywords: winding number
zbMATH Keywords: normal solvability
author: Andrei A. Lyashenko
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
cites work: On generalized eigenfunctions of an operator which is related to a problem of S. L. Sobolev
cites work: Q3717469
cites work: Q5825924
cites work: Q3218514
cites work: The Dirichlet Problem for a Hyperbolic Equation
description: scientific article; zbMATH DE number 130810