KohlerJobin meets Ehrhard: the sharp lower bound for the Gaussian principal frequency while the Gaussian torsional rigidity is fixed, via rearrangements
Abstract
In this note, we provide an adaptation of the KohlerJobin rearrangement technique to the setting of the Gauss space. As a result, we prove the Gaussian analogue of the KohlerJobin's resolution of a conjecture of PólyaSzegö: when the Gaussian torsional rigidity of a (convex) domain is fixed, the Gaussian principal frequency is minimized for the halfspace. At the core of this rearrangement technique is the idea of considering a "modified" torsional rigidity, with respect to a given function, and rearranging its layers to halfspaces, in a particular way; the Rayleigh quotient decreases with this procedure. We emphasize that the analogy of the Gaussian case with the Lebesgue case is not to be expected here, as in addition to some soft symmetrization ideas, the argument relies on the properties of some special functions; the fact that this analogy does hold is somewhat of a miracle.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 arXiv:
 arXiv:2201.06191
 Bibcode:
 2022arXiv220106191H
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry;
 Mathematics  Metric Geometry;
 Mathematics  Probability;
 46F20;
 49R05;
 52A40
 EPrint:
 15 pages