A Weighted Sobolev Space Theory of Parabolic Stochastic PDEs on Non-smooth Domains

Abstract

In this article, we study parabolic stochastic partial differential equations (see Eq. (1.1)) defined on arbitrary bounded domain admitting the Hardy inequality integral(O)vertical bar rho(-1)g vertical bar(2) dx <= c integral(O) vertical bar gx vertical bar(2)dx, for all g is an element of c(0)(infinity) (O), where rho(x) = dist(x, partial derivative(O)). Existence and uniqueness results are given in weighted Sobolev spaces h(p,theta)(gamma) (O, T), where rho is an element of [2, infinity), gamma is an element of R is the number of derivatives of solutions and theta controls the boundary behavior of solutions (see Definition 2.5). Furthermore, several Holder estimates of the solutions are also obtained. It is allowed that the coefficients of the equations blow up near the boundary.