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 $\mu$. 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 $\Omega =G^{-1}(-\infty ,0]$. 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} λu-L_{ν,Ω} 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 \[(ψ,ϕ)\mapsto \int_Ω\langle\nabla_Hψ,\nabla_Hϕ\rangle_Hdν\qquadψ,ϕ\in W^{1,2}(Ω,ν).\] In addition we will show that if $u$ is a weak solution of problem $\lambda u-L_{\nu ,\Omega }u=f$, 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.

References in corpus (4)

• Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space
• Dirichlet spaces on H-convex sets in Wiener space
• Maximal Sobolev regularity for solutions of elliptic equations in infinite dimensional Banach spaces endowed with a weighted Gaussian measure
• Sobolev spaces with respect to a weighted Gaussian measures in infinite dimensions

Cited by in corpus (10)

• Maximal Sobolev regularity for solutions of elliptic equations in infinite dimensional Banach spaces endowed with a weighted Gaussian measure
• Schauder theorems for a class of (pseudo-)differential operators on finite and infinite dimensional state spaces
• Regularizing properties of (non-Gaussian) transition semigroups in Hilbert spaces
• Gradient estimates for perturbed Ornstein-Uhlenbeck semigroups on infinite dimensional convex domains
• Sobolev spaces with respect to a weighted Gaussian measures in infinite dimensions
• On the domain of elliptic operators defined in subsets of Wiener spaces
• On functions of bounded variation on convex domains in Hilbert spaces
• Differentiability in infinite dimension and the Malliavin calculus
• Regarding the domain of non-symmetric and, possibly, degenerate Ornstein--Uhlenbeck operators in separable Banach spaces
• $L^2$-theory for transitions semigroups associated to dissipative systems