A nonlocal approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties

TL;DR

This paper investigates a nonlocal approximation of the Gaussian perimeter, establishing Gamma convergence to the local perimeter, and reveals unique properties of stationary points and symmetrization in this nonlocal setting.

Contribution

It introduces a nonlocal Gaussian perimeter approximation, proves its Gamma convergence, and uncovers novel properties of stationary points and symmetrization effects.

Findings

Gamma convergence to the local Gaussian perimeter

Halfspaces are stationary points only when boundary passes through the origin

Ehrhard symmetrization can increase the nonlocal Gaussian perimeter

Abstract

We study a non local approximation of the Gaussian perimeter, proving the Gamma convergence to the local one. Surprisingly, in contrast with the local setting, the halfspace turns out to be a volume constrained stationary point if and only if the boundary hyperplane passes through the origin. In particular, this implies that Ehrhard symmetrization can in general increase the considered non local Gaussian perimeter.