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

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 $\kappa_0$.