Japan Journal of Industrial and Applied Mathematics

Computer-Assisted Proofs for Semilinear Elliptic Boundary Value Problems

Michael Plum

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Abstract

For second-order semilinear elliptic boundary value problems on bounded or unbounded domains, a general computer-assisted method for proving the existence of a solution in a ``close'' and explicit neighborhood of an approximate solution, computed by numerical means, is proposed. To achieve such an existence and enclosure result, we apply Banach's fixed-point theorem to an equivalent problem for the error, i.e., the difference between exact and approximate solution. The verification of the conditions posed for the fixed-point argument requires various analytical and numerical techniques, for example the computation of eigenvalue bounds for the linearization at the approximate solution. The method is used to prove existence and multiplicity results for some specific examples.

Article information

Source
Japan J. Indust. Appl. Math. Volume 26, Number 2-3 (2009), 419-442.

Dates
First available in Project Euclid: 1 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.jjiam/1265033789

Mathematical Reviews number (MathSciNet)
MR2589483

Zentralblatt MATH identifier
1186.35073

Keywords
elliptic boundary value problem semilinear computer-assisted proof existence enclosures error bounds multiplicity

Citation

Plum, Michael. Computer-Assisted Proofs for Semilinear Elliptic Boundary Value Problems. Japan J. Indust. Appl. Math. 26 (2009), no. 2-3, 419--442.https://projecteuclid.org/euclid.jjiam/1265033789


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