Maximal [formula omitted] regularity for Dirichlet problems in Hilbert spaces

Abstract

We consider the Dirichlet problem λU−LU=F in O, U=0 on ∂O. Here F∈L2(O,μ) where μ is a nondegenerate centered Gaussian measure in a Hilbert space X, L is an Ornstein–Uhlenbeck operator, and O is an open set in X with good boundary. We address the problem whether the weak solution U belongs to the Sobolev space W2,2(O,μ). It is well known that the question has positive answer if O=X; if O≠X we give a sufficient condition in terms of geometric properties of the boundary ∂O.