Constructive Aspects of the Dirichlet Problem

Douglas Bridges; Wang Yuchuan

doi:10.3217/jucs-003-11-1148

JUCS - Journal of Universal Computer Science 3(11): 1148-1161, doi: 10.3217/jucs-003-11-1148

Constructive Aspects of the Dirichlet Problem

Douglas S. Bridges^‡, Wang Yuchuan^§

‡ Department of Mathematics, University of Waikato, Hamilton, New Zealand§ Department of Mathematics, University of Waikato, New Zealand

Corresponding author: Douglas Bridges ( douglas@waikato.ac.nz )

This article is freely available under the J.UCS Open Content License.

Citation: Bridges DS, Yuchuan W (1997) Constructive Aspects of the Dirichlet Problem. JUCS - Journal of Universal Computer Science 3(11): 1148-1161. https://doi.org/10.3217/jucs-003-11-1148

Abstract

We examine, within the framework of Bishop's constructive mathematics, various classical methods for proving the existence of weak solutions of the Dirichlet Problem, with a view to showing why those methods do not immediately translate into viable constructive ones. In particular, we discuss the equivalence of the existence of weak solutions of the Dirichlet Problem and the existence of minimizers for certain associated integral functionals. Our analysis pinpoints exactly what is needed to find weak solutions of the Dirichlet Problem: namely, the computation of either the norm of a linear functional on a certain Hilbert space or, equivalently, the infimum of an associated integral functional. 1.) Proceedings of the First Japan-New Zealand Workshop on Logic in Computer Science, special issue editors D.S. Bridges, C.S. Calude, M.J. Dinneen and B. Khoussainov.

Keywords

Dirichlet problem, Hilbert space, Bishop's constructive mathematics