A class of non-cylindrical domains for parabolic equations

TL;DR

This paper introduces a new class of non-cylindrical domains with mixed Lipschitz boundary regularity, enabling the formulation and solution of parabolic PDEs like the heat equation within these domains.

Contribution

It defines a novel class of non-cylindrical domains with mixed Lipschitz regularity and adapts the implicit function theorem to handle these domains for parabolic equations.

Findings

The class of domains is comparable to previously studied types.

The adapted implicit function theorem is effective for these domains.

Existence and solvability of Dirichlet problems are established.

Abstract

We present a class of non-cylindrical domains where Dirichlet-type problems for parabolic equations, such as the heat equation, can be posed and solved. The regularity for the boundary of this class of domains is a mixed Lipschitz condition, as described in the bulk of the paper. The main tool is an adequate version of the implicit function theorem for functions with this kind of regularity. It is proved that the class introduced herein is of the same type as domains previously considered by several authors.