Calculus of Variations and Geometric Measure Theory

M. Barchiesi - V. Julin

Symmetry of minimizers of a Gaussian isoperimetric problem

created by barchiesi on 08 May 2018
modified on 16 Mar 2021


Published Paper

Inserted: 8 may 2018
Last Updated: 16 mar 2021

Journal: Probab. Theory Related Fields
Volume: 177
Pages: 217-256
Year: 2020
Doi: 10.1007/s00440-019-00947-9


We study an isoperimetric problem described by a functional that consists of the standard Gaussian perimeter and the norm of the barycenter. This second term has a repulsive effect, and it is in competition with the perimeter. Because of that, in general the solution is not the half-space. We characterize all the minimizers of this functional, when the volume is close to one, by proving that the minimizer is either the half-space or the symmetric strip, depending on the strength of the repulsive term. As a corollary, we obtain that the symmetric strip is the solution of the Gaussian isoperimetric problem among symmetric sets when the volume is close to one.

Keywords: quantitative isoperimetric inequality, symmetric solutions