On a reverse Kohler-Jobin inequality

TL;DR

This paper investigates shape optimization problems involving the product of eigenvalues and torsional rigidity, establishing existence of maximizers for large q and identifying nearly spherical domains as optimal.

Contribution

It proves the existence of maximizers among quasi-open sets for large q and characterizes nearly spherical domains as optimal for the maximization problem.

Findings

Existence of maximizers for large q among quasi-open sets.

The ball is optimal among nearly spherical domains.

Complete characterization of the infimum of the product.

Abstract

We consider the shape optimization problems for the quantities $\lambda(\Omega)T^q(\Omega)$, where $\Omega$ varies among open sets of $\mathbb{R}^d$ with a prescribed Lebesgue measure. While the characterization of the infimum is completely clear, the same does not happen for the maximization in the case $q>1$. We prove that for $q$ large enough a maximizing domain exists among quasi-open sets and that the ball is optimal among {\it nearly spherical domains}.