On Polya's inequality for torsional rigidity and first Dirichlet eigenvalue

TL;DR

This paper investigates properties of a set function involving torsional rigidity and the first Dirichlet eigenvalue, improving classical bounds and constructing sets that nearly attain the bound in Euclidean spaces.

Contribution

It improves Polya's inequality for the combined measure of torsional rigidity and eigenvalue, providing a sharper upper bound and explicit near-extremal examples.

Findings

Improved the classical Polya bound for the set function F.

Derived a new upper bound involving the volume and torsional rigidity.

Constructed sets that nearly attain the bound for any dimension and epsilon.

Abstract

Let $\Omega$ be an open set in Euclidean space with finite Lebesgue measure $|\Omega|$. We obtain some properties of the set function $F:\Omega\mapsto \R^+$ defined by $$ F(\Omega)=\frac{T(\Omega)\lambda_1(\Omega)}{|\Omega|} ,$$ where $T(\Omega)$ and $\lambda_1(\Omega)$ are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical P\'olya bound $F(\Omega)\le 1,$ and show that $$F(\Omega)\le 1- \nu_m T(\Omega)|\Omega|^{-1-\frac2m},$$ where $\nu_m$ depends only on $m$. For any $m=2,3,\dots$ and $\epsilon\in (0,1)$ we construct an open set $\Omega_{\epsilon}\subset \R^m$ such that $F(\Omega_{\epsilon})\ge 1-\epsilon$.