An $L_p$-theory for the stochastic heat equation on angular domains in $\mathbb{R}^2$ with mixed weights

TL;DR

This paper develops a refined $L_p$-theory for the stochastic heat equation on angular domains in $R^2$, accounting for boundary and vertex singularities through mixed weights, and establishes higher order regularity results.

Contribution

It introduces a new $L_p$-estimate framework with mixed weights for the stochastic heat equation on angular domains, capturing singularities from boundary and vertex effects.

Findings

Established refined $L_p$-estimates with mixed weights

Proved higher order $L_p$-Sobolev regularity

Captured singularities due to boundary and vertex effects

Abstract

We establish a refined $L_p$-estimate ($p\geq 2$) for the stochastic heat equation on angular domains in $\mathbb{R}^2$ with mixed weights based on both, the distance to the boundary and the distance to the vertex. This way we can capture both causes for singularities of the solution: the incompatibility of noise and boundary condition on the one hand and the influence of boundary singularities (here, the vertex) on the other hand. Higher order $L_p$-Sobolev regularity with mixed weights is also established.