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

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

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