Open Access
October 2009 Computer-Assisted Proofs for Semilinear Elliptic Boundary Value Problems
Michael Plum
Japan J. Indust. Appl. Math. 26(2-3): 419-442 (October 2009).


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.


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Michael Plum. "Computer-Assisted Proofs for Semilinear Elliptic Boundary Value Problems." Japan J. Indust. Appl. Math. 26 (2-3) 419 - 442, October 2009.


Published: October 2009
First available in Project Euclid: 1 February 2010

zbMATH: 1186.35073
MathSciNet: MR2589483

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

Rights: Copyright © 2009 The Japan Society for Industrial and Applied Mathematics

Vol.26 • No. 2-3 • October 2009
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