An $L_p$-estimate for the stochastic heat equation on an angular domain in $\mathbb{R}^2$

TL;DR

This paper establishes a weighted $L_p$-estimate for the stochastic heat equation on a planar angular domain, enabling proof of existence and uniqueness of solutions with singular behavior near the vertex.

Contribution

It provides the first weighted $L_p$-estimate for the stochastic heat equation on angular domains, accounting for boundary singularities and explicitly relating weights to the domain's angle.

Findings

Weighted $L_p$-estimate derived for stochastic convolution

Existence and uniqueness of solutions proved in weighted Sobolev spaces

Explicit relation between weights and domain angle established

Abstract

We prove a weighted $L_p$-estimate for the stochastic convolution associated to the stochastic heat equation with zero Dirichlet boundary condition on a planar angular domain $\mathcal{D}_{\kappa_0}\subset\mathbb{R}^2$ with angle $\kappa_0\in(0,2\pi)$. Furthermore, we use this estimate to establish existence and uniqueness of a solution to the corresponding equation in suitable weighted $L_p$-Sobolev spaces. In order to capture the singular behaviour of the solution and its derivatives at the vertex, we use powers of the distance to the vertex as weight functions. The admissible range of weight parameters depends explicitly on the angle $\kappa_0$.