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

Abstract

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\hat{I}¼$. 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 $\hat{I}½=e^{-U}\hat{I}¼$ and $\hat{I}\text{c◯}=G^{-1}(-\infty ,0]$. 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} λu-L_{ν,Ω} u=f, \end{align} where $\hat{I}»>0$, $f\in L^2(\hat{I}\text{c◯},\hat{I}½)$ and $L_{\hat{I}½,\hat{I}\text{c◯}}$ is the self-adjoint operator associated with the quadratic form \[(ψ,ϕ)\mapsto \int_Ω\langle\nabla_Hψ,\nabla_Hϕ\rangle_Hdν\qquadψ,ϕ\in W^{1,2}(Ω,ν).\] In addition we will show that if $u$ is a weak solution of problem $\hat{I}»u-L_{\hat{I}½,\hat{I}\text{c◯}}u=f$, with $\hat{I}»>0$ and $f\in L^2(\hat{I}\text{c◯},\hat{I}½)$, then it satisfies a Neumann type condition at the boundary, namely for $\ddot{I}$-a.e. $x\in G^{-1}(0)$ \[\left\langle\,\text{Tr}\,(\nabla_Hu)(x),\,\text{Tr}\,(\nabla_H G)(x)\right\rangle_H=0,\] where $\ddot{I}$ is the Feyel--de La Pradelle Hausdorff--Gauss surface measure and $\text{Tr}$ is the trace operator.