Maximal Sobolev regularity for solutions of elliptic equations in Banach spaces endowed with a weighted Gaussian measure: the convex subset case

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron–Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\to \mathbb{R}$ and $G:X\to \mathbb{R}$. We let $\nu =e^{-U}\mu$ and $\mathrm{\Omega }=G^{-1}(-\mathrm{\infty },0]$. In this paper we are interested in the $W^{2,2}$ regularity of the weak solutions of elliptic equations of the type \begin{align}\label{Probelma in abstract} \lambda u-L_{\nu,\Omega} u=f, \end{align} where $\lambda >0$, $f\in L^2(\mathrm{\Omega },\nu )$ and $L_{\nu ,\mathrm{\Omega }}$ is the self-adjoint operator associated with the quadratic form $$(\psi ,\varphi )\mapsto {\int }_{\mathrm{\Omega }}⟨{\mathrm{\nabla }}_H\psi ,{\mathrm{\nabla }}_H\varphi {⟩}_{H}d\nu \psi ,\varphi \in W^{1,2}(\mathrm{\Omega },\nu )\mathrm{.}$$ In addition we will show that if $u$ is a weak solution of problem $\lambda u-L_{\nu ,\mathrm{\Omega }}u=f$, with $\lambda >0$ and $f\in L^2(\Omega,\nu)$, then it satisfies a Neumann type condition at the boundary, namely for $\rho$-a.e. $x\in G^{-1}(0)$ $${⟨\text{Tr}({\mathrm{\nabla }}_{H}u)(x),\text{Tr}({\mathrm{\nabla }}_{H}G)(x)⟩}_H=0,$$ where $\rho$ is the Feyel–de La Pradelle Hausdorff–Gauss surface measure and $\text{Tr}$ is the trace operator.