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Isoperimetric and stable sets for log-concave perturbations of Gaussian measures. (English) Zbl 1304.49096

Summary: 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 the 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.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49Q10 Optimization of shapes other than minimal surfaces
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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