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

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.