Isoperimetric and Stable Sets for Log-ConcavePerturbations of Gaussian Measures

Abstract

AbstractLet be an open half-space or slab in ℝn+1endowed with a perturbation of the Gaussian measureof the form f (p) := exp(ω(p) − c|p|2), where c > 0 and ω is a smooth concave function depending only onthe signed distance from the linear hyperplane parallel to ∂ Ω. In this work we follow a variational approach to show that half-spaces perpendicular to ∂ Ω 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-spacesparallel or perpendicular to ∂ Ω as the unique stable sets with small singular set and null weighted capacity.Our methods also apply for = ℝn+1, which produces in particular the classification of stable sets in Gaussspace and a new proof of the Gaussian isoperimetric inequality. Finally, we use optimal transport to studythe weighted minimizers when the perturbation term ω is concave and possibly non-smooth.