Analysis Seminar

Kohler-Jobin Meets Ehrhard: the Sharp Lower Bound for the Gaussian Principal Frequency while the Gaussian Torsional Rigidity is Fixed

Orli Herscovici, Georgia Institute of Technology

Friday, March 25th, 2022
1:00 PM - 2:00 PM
Hylan 1106A

In this talk, we show an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we present the Gaussian analogue of the Kohler-Jobin’s resolution of a conjecture of Polya-Szego: when the Gaussian torsional rigidity of a (convex) domain is fixed, the Gaussian principal frequency is minimized for the half-space. 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 half-spaces, 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.

Based on joint work with Galyna Livshyts.

Event contact: dan dot geba at rochester dot edu