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

M. van den Berg, V. Ferone, C. Nitsch, C. Trombetti

Published 2016-02-15Version 1

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 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 on $m$ only. For any $\epsilon\in (0,1)$ we construct an open set $\Omega_{\epsilon}$ such that $F(\Omega_{\epsilon})\ge 1-\epsilon$.