A weighted Sobolev space theory of parabolic stochastic PDEs on non-smooth domains

TL;DR

This paper develops a weighted Sobolev space framework for analyzing parabolic stochastic PDEs on arbitrary non-smooth domains, allowing boundary behavior and coefficient blow-up, with existence, uniqueness, and regularity results.

Contribution

It introduces a novel weighted Sobolev space theory for stochastic PDEs on irregular domains, accommodating boundary singularities and unbounded coefficients.

Findings

Existence and uniqueness of solutions in weighted Sobolev spaces.

Hölder regularity estimates for solutions.

Framework accommodates boundary singularities and coefficient blow-up.

Abstract

In this paper we study parabolic stochastic partial differential equations defined on arbitrary bounded domain $\cO \subset \bR^d$ allowing Hardy inequality: $$ \int_{\cO}|\rho^{-1}g|^2\,dx\leq C\int_{\cO}|g_x|^2 dx, \quad \forall g\in C^{\infty}_0(\cO), $$ where $\rho(x)=\text{dist}(x,\partial \cO)$. Existence and uniqueness results are given in weighted Sobolev spaces $\frH^{\gamma}_{p,\theta}(\cO,T)$, where $p\in [2,\infty)$, $\gamma\in \bR$ is the number of derivatives of solutions and $\theta$ controls the boundary behavior of solutions. Furthermore several H\"older estimates of the solutions are also obtained. It is allowed that the coefficients of the equations blow up near the boundary.