Approximation and relaxation of perimeter in the Wiener space

TL;DR

This paper characterizes the relaxed perimeter in infinite-dimensional Wiener space and demonstrates that rescaled Allen-Cahn functionals approximate this relaxation through Gamma-convergence.

Contribution

It introduces a novel characterization of perimeter relaxation in Wiener space and links it to Allen-Cahn functionals via Gamma-convergence.

Findings

Relaxed perimeter characterized in Wiener space

Rescaled Allen-Cahn functionals approximate the relaxed perimeter

Gamma-convergence established between functionals

Abstract

We characterize the relaxation of the perimeter in an infinite dimensional Wiener space, with respect to the weak L^2-topology. We also show that the rescaled Allen-Cahn functionals approximate this relaxed functional in the sense of Gamma-convergence.