Maximal $L^2$ regularity for Ornstein-Uhlenbeck equation in convex sets of Banach spaces

Abstract

We study the elliptic equation $λu-L^Ωu=f$ 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 $f\in L^2(Ω,γ)$ 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.