On the Dirichlet problem for the nonlinear equation of the vibrating string. I
type: Item
instance of: scholarly article
title: On the Dirichlet problem for the nonlinear equation of the vibrating string. I
zbMATH Open document ID: 0797.35124
published in: Journal of Mathematics of Kyoto University
publication date: 1994-10-27T00:00:00Z
review text: The aim of this paper is the solvability of the Dirichlet problem \[ u_{xx} -u_{yy} + f(x,y,u) = 0,\;(x,y) \in \Omega,\quad u(x,y) = 0,\;(x,y) \in \partial \Omega, \tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^ 2\), convex relative to the characteristics \(x \pm y = cte\). The existence and uniqueness of weak solutions of (1) is proved, for some class of domains \(\Omega\) when \(f\) is continuous, monotone with respect to \(u\) and satisfies an additional growth restriction related to the first negative and first positive eigenvalue of the operator \(A\), the closure in \(L^ 2(\Omega)\) of the symmetric operator \(A_ 0u = u_{xy}\), \(D(A_ 0) = C^ \infty (\overline \Omega) \cap W_ 0^{1,2} (\Omega)\). Also, some regularity properties of the solutions are studied.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 558904
zbMATH Keywords: nonlinear vibrating string equation
zbMATH Keywords: special domains
zbMATH Keywords: normal solvability
zbMATH Keywords: Dirichlet problem
author: Andrei A. Lyashenko
reviewed by: Antonio Cañada
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519272
OpenAlex ID: W4252164623
DOI: 10.1215/KJM/1250519272
description: scientific article; zbMATH DE number 558904