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

We study the elliptic equation $\lambda u-L^{\mathrm{\Omega }}u=f$ in an open convex subset $\mathrm{\Omega }$ of an infinite dimensional separable Banach space $X$ endowed with a centered non-degenerate Gaussian measure $\gamma$, where $L^{\mathrm{\Omega }}$ is the Ornstein-Uhlenbeck operator. We prove that for $\lambda >0$ and $f\in L^2(\Omega,\gamma)$theweaksolution$u$belongstotheSobolevspace$W^{2,2}(\Omega,\gamma)$\mathrm{.}Moreoverweprovethat$u$ satisfies the Neumann boundary condition in the sense of traces at the boundary of $\mathrm{\Omega }$. This is done by finite dimensional approximation.