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

Abstract

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