Maximal L2 regularity for Ornstein–Uhlenbeck equation in convex sets of Banach spaces

We study the elliptic equation λu−LΩu=fλu−LΩu=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 LΩLΩ is the Ornstein–Uhlenbeck operator. We prove that for λ>0λ>0 and f∈L2(Ω,γ)f∈L2(Ω,γ) the weak solution u   belongs to the Sobolev space W2,2(Ω,γ)W2,2(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.