Sobolev space theory and Holder estimates for the stochastic partial differential equations on conic and polygonal domains

Abstract

We establish existence, uniqueness, and Sobolev and Holder regularity results for the stochastic partial differential equation du = sigma(d)(i,j=1) a(ij)u(xixj) + f(0) + sigma(d)(i=1)f(xi)(i))dt + sigma(infinity)(k=1)g(k)dw(tk) , t > 0, x is an element of D given with non-zero initial data. Here {w(t)(k) : k = 1, 2, middotmiddotmiddot} is a family of independent Wiener processes defined on a probability space (omega, P), a(ij) = a(ij)(omega, t) are merely measurable functions on omega x (0, infinity), and D is either a polygonal domain in R-2 or an arbitrary dimensional conic domain of the type D(M) := {x is an element of R-d : x/|x| is an element of M}, M ? Sd-1, (d >= 2) (0.1) where M is an open subset of Sd-1 with C-2 boundary. We measure the Sobolev and Holder regularities of arbitrary order derivatives of the solution using a system of mixed weights consisting of appropriate powers of the distance to the vertices and of the distance to the boundary. The ranges of admissible powers of the distance to the vertices and to the boundary are sharp. (C) 2022 Elsevier Inc. All rights reserved.