Isoperimetric and stable sets for log-concave perturbations of Gaussian measures

Abstract

Let $\Omega$ be an open half-space or slab in ${\mathbb{R}}^{n+1}$ endowed with a perturbation of the Gaussian measure of the form $f(p):=\exp (\omega (p)-c|p{|}^2)$, where $c>0$ and $\omega$ is a smooth concave function depending only on the signed distance from the linear hyperplane parallel to $\partial \Omega$. In this work we follow a variational approach to show that half-spaces perpendicular to $\partial \Omega$ uniquely minimize the weighted perimeter in $\Omega$ among sets enclosing the same weighted volume. The main ingredient of the proof is the characterization of half-spaces parallel or perpendicular to $\partial \Omega$ as the unique stable sets with small singular set and null weighted capacity. Our methods also apply for $\Omega ={\mathbb{R}}^{n+1}$, which produces in particular the classification of stable sets in Gauss space and a new proof of the Gaussian isoperimetric inequality. Finally, we use optimal transport to study the weighted minimizers when the perturbation term $\omega$ is concave and possibly non-smooth.

arXiv commentfinal version, to appear in Analysis and Geometry in Metric Spaces