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

Research paper by **G. Cappa, S. Ferrari**

Indexed on: **23 Sep '16**Published on: **23 Sep '16**Published in: **arXiv - Mathematics - Analysis of PDEs**

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#### 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.