On the regularity of the stochastic heat equation on polygonal domains in $R^2$

TL;DR

This paper proves existence, uniqueness, and detailed regularity results for the stochastic heat equation on polygonal domains in 2D, accounting for boundary and vertex singularities using weighted Sobolev spaces.

Contribution

It introduces a novel weighted Sobolev regularity framework that captures boundary and vertex singularities for stochastic heat equations on polygonal domains.

Findings

Established existence and uniqueness of solutions.

Derived higher order weighted Sobolev regularity.

Characterized sharp admissible weights based on interior angles.

Abstract

We establish existence, uniqueness and higher order weighted $L_p$-Sobolev regularity for the stochastic heat equation with zero Dirichlet boundary condition on angular domains and on polygonal domains in $\mathbb{R}^2$. We use a system of mixed weights consisting of appropriate powers of the distance to the vertexes and of the distance to the boundary to measure the regularity with respect to the space variable. In this way we can capture the influence of both main sources for singularities: the incompatibility between noise and boundary condition on the one hand and the singularities of the boundary on the other hand. The range of admissible powers of the distance to the vertexes is described in terms of the maximal interior angle and is sharp.