Isoperimetric and Stable Sets for Log-Concave Perturbations of Gaussian Measures

Rosales, César

Analysis and Geometry in Metric Spaces, Tome 2 (2014) no. 1

Résumé

Let be an open half-space or slab in Rn+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 ∂ 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-spaces parallel 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 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.

Détail

Mots-clés : Log-concave densities, Gaussian measures, isoperimetric problems, stable sets, free boundary hypersurfaces