Maximal Sobolev regularity for solutions of elliptic equations in infinite dimensional Banach spaces endowed with a weighted Gaussian measure

TL;DR

This paper establishes maximal Sobolev regularity for solutions of elliptic equations in infinite-dimensional Banach spaces with weighted Gaussian measures, extending regularity results to a broad class of weighted measures.

Contribution

It proves $W^{2,2}$ regularity for solutions of elliptic equations in infinite-dimensional Banach spaces with a weighted Gaussian measure, under broad conditions.

Findings

Proves $W^{2,2}$ regularity for elliptic solutions in infinite dimensions.

Extends regularity results to weighted Gaussian measures with convex potentials.

Provides a framework for analyzing elliptic equations in 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 $H$. Let $\nu=e^{-U}\mu$, where $e^{-U}$ is a sufficiently regular weight and $U:X\rightarrow\mathbb{R}$ is a convex and continuous function. In this paper we are interested in the $W^{2,2}$ regularity of the weak solutions of elliptic equations of the type \[\lambda u-L_\nu u=f,\] where $\lambda>0$, $f\in L^2(X,\nu)$ and $L_\nu$ is the self-adjoint operator associated with the quadratic form \[(\psi,\varphi)\mapsto \int_X\left\langle\nabla_H\psi,\nabla_H\varphi\right\rangle_Hd\nu\qquad\psi,\varphi\in W^{1,2}(X,\nu).\]