A remark on a conjecture on the symmetric Gaussian Problem
Abstract
In this paper we study the functional given by the integral of the mean curvature of a convex set with Gaussian weight with Gaussian volume constraint. It was conjectured that the ball centered at the origin is the only minimizer of such a functional for certain value of the mass. We give a positive answer in dimension two while in higher dimension the situation is different. In fact, for small value of mass the ball centered at the origin is a local minimizer while for large values the ball is a maximizer among convex sets with uniform bound on the curvature.
 Publication:

arXiv eprints
 Pub Date:
 June 2023
 DOI:
 10.48550/arXiv.2306.04522
 arXiv:
 arXiv:2306.04522
 Bibcode:
 2023arXiv230604522F
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 Proceedings of the Edinburgh Mathematical Society 67 (2024) 643661