Maximal L 2 regularity for Ornstein–Uhlenbeck equation in convex sets of Banach spaces

Abstract We study the elliptic equation λ u − L Ω u = f in an open convex subset Ω of an infinite dimensional separable Banach space X endowed with a centered non-degenerate Gaussian measure γ , where L Ω is the Ornstein–Uhlenbeck operator. We prove that for λ > 0 and f ∈ L 2 ( Ω , γ ) the weak solution u belongs to the Sobolev space W 2 , 2 ( Ω , γ ) . Moreover we prove that u satisfies the Neumann boundary condition in the sense of traces at the boundary of Ω. This is done by finite dimensional approximation.