On the Perron solution of the caloric Dirichlet problem: an elementary approach

TL;DR

The paper introduces an elementary method to construct Perron solutions for the caloric Dirichlet problem on a broad class of domains, simplifying the approach to solving heat equation boundary value problems.

Contribution

It presents a new, elementary approach to construct Perron solutions for the caloric Dirichlet problem on almost regular domains, expanding the class of solvable problems.

Findings

Constructed a basis of almost regular domains for the caloric Dirichlet problem.

Provided an elementary procedure to build Perron solutions on any bounded domain.

Extended the applicability of Perron method to a wider class of domains.

Abstract

By an easy trick taken from caloric polynomial theory we construct a family $\mathscr{B}$ of $almost\ regular$ domains for the caloric Dirichlet problem. $\mathscr{B}$ is a basis of the Euclidean topology. This allows to build, with a basically elementary procedure, the Perron solution to the caloric Dirichlet problem on every bounded domain.