Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems
type: Item
instance of: scholarly article
title: Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems
zbMATH Open document ID: 0771.65087
DOI: 10.1007/BF02238648
published in: Computing
publication date: 1993-01-13T00:00:00Z
review text: A computer-assisted method for proving the existence of a solution of the problem \(-\Delta U(x)+F(x,U(x),U_ x(x))=0\) \((x\in\Omega)\), \(B[U](x)=0\) \((x\in\partial\Omega)\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\) \((n=2\) or 3) and the nonlinearity \(F\) is defined on \(\overline\Omega\times\mathbb{R}\times\mathbb{R}^ n\) with values \(F(x,y,z)\in\mathbb{R}\), is presented. It is assumed that \(F\) grows at most quadratically with respect to the gradient \(U_ x\) and with a mixed-type linear boundary operator \(B\). All kinds of monotonicity conditions on the nonlinearity \(F\), or inverse-positivity assumptions are avoided. Some numerical examples are reported finally.
reviewed by: K.Zlateva
Mathematics Subject Classification ID: 65Z05
Mathematics Subject Classification ID: 68T15
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 65G30
zbMATH DE Number: 86032
zbMATH Keywords: explicit Sobolev imbeddings
zbMATH Keywords: eigenvalue bounds
zbMATH Keywords: enclosure
zbMATH Keywords: numerical existence proofs
zbMATH Keywords: explicit bounds for solutions
zbMATH Keywords: nonlinear elliptic boundary value problems
zbMATH Keywords: computer-assisted method
zbMATH Keywords: existence
zbMATH Keywords: numerical examples
author: Michael Plum
describes a project that uses: ACRITH-XSC
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02238648
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description: scientific article; zbMATH DE number 86032