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

TL;DR

This paper establishes maximal Sobolev regularity for solutions of elliptic equations in Banach spaces with weighted Gaussian measures, focusing on convex subsets and boundary conditions.

Contribution

It proves $W^{2,2}$ regularity for weak solutions of elliptic equations in Banach spaces with Gaussian measures, including boundary Neumann conditions.

Findings

Proves $W^{2,2}$ regularity for solutions.

Shows boundary Neumann condition at the convex subset boundary.

Extends regularity results to infinite-dimensional Banach spaces.

Abstract

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 $U:X\rightarrow\mathbb{R}$ and $G:X\rightarrow \mathbb{R}$. We let $\nu=e^{-U}\mu$ and $\Omega=G^{-1}(-\infty,0]$. 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} \lambda u-L_{\nu,\Omega} u=f, \end{align} where $\lambda>0$, $f\in L^2(\Omega,\nu)$ and $L_{\nu,\Omega}$ is the self-adjoint operator associated with the quadratic form \[(\psi,\phi)\mapsto \int_\Omega\langle\nabla_H\psi,\nabla_H\phi\rangle_Hd\nu\qquad\psi,\phi\in W^{1,2}(\Omega,\nu).\] In addition we will show that if $u$ is a weak solution of problem $\lambda u-L_{\nu,\Omega} u=f$, 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. $x\in G^{-1}(0)$ \[\left\langle\,\text{Tr}\,(\nabla_Hu)(x),\,\text{Tr}\,(\nabla_H G)(x)\right\rangle_H=0,\] where $\rho$ is the Feyel--de La Pradelle Hausdorff--Gauss surface measure and $\text{Tr}$ is the trace operator.