Calculus of Variations and Geometric Measure Theory

D. Addona - G. Menegatti - M. Miranda Jr

Gradient contractivity of a rescaled resolvent on domains in Wiener spaces

created by addona on 22 Apr 2024

[BibTeX]

preprint

Inserted: 22 apr 2024
Last Updated: 22 apr 2024

Year: 2024

ArXiv: 2404.10611 PDF

Abstract:

Given an abstract Wiener space $(X,\gamma,H)$, we consider an open set $O\subseteq X$ which satisfies certain smoothness and mean-curvature conditions. We prove that the rescaled resolvent operator associated to the Ornstein-Uhlenbeck operator with homogeneous Dirichlet boundary conditions on $O$ is gradient contractive in $L^p(X,\gamma)$ for every $p\in(1,\infty)$. This is the Gaussian counterpart of an analogous result for the rescaled resolvent operator associated to the Laplace operator $\Delta$ in $L^p$ with respect to the Lebesgue measure, $p\in[1,\infty)$, with homogeneous Dirichlet boundary conditions on a bounded convex open set $O\subseteq \mathbb R^n$.


Download: