$W^{2,p}$-solutions of parabolic SPDEs in general domains

TL;DR

This paper establishes existence, uniqueness, and regularity results for solutions to a class of stochastic parabolic PDEs in general domains within Sobolev spaces, extending the understanding of their analytical properties.

Contribution

It provides new existence and regularity results for $W^{2,p}$-solutions of stochastic parabolic PDEs in arbitrary domains, under specific compatibility conditions.

Findings

Proved existence and uniqueness of solutions in Sobolev spaces.

Established Hölder continuity of solutions and derivatives.

Extended regularity theory to general domains for stochastic PDEs.

Abstract

The Dirichlet problem for a class of stochastic partial differential equations is studied in Sobolev spaces. The existence and uniqueness result is proved under certain compatibility conditions that ensure the finiteness of $L^{p}(\Omega\times(0,T),W^{2,p}(G))$-norms of solutions. The H\"older continuity of solutions and their derivatives is also obtained by embedding.