Kohler-Jobin meets Ehrhard: the sharp lower bound for the Gaussian principal frequency while the Gaussian torsional rigidity is fixed, via rearrangements

TL;DR

This paper adapts the Kohler-Jobin rearrangement technique to Gaussian space, proving that among convex domains with fixed Gaussian torsional rigidity, the half-space minimizes the Gaussian principal frequency.

Contribution

It establishes the Gaussian analogue of Kohler-Jobin's resolution of a Pólya-Szegö conjecture using a novel rearrangement method involving modified torsional rigidity.

Findings

Gaussian principal frequency minimized by half-space

Rearrangement decreases Rayleigh quotient

Extension of Kohler-Jobin technique to Gaussian space

Abstract

In this note, we provide an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we prove the Gaussian analogue of the Kohler-Jobin's resolution of a conjecture of P\'{o}lya-Szeg\"o: 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.