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

Abstract

Let $\Omega$ be an open set in Euclidean space with finite Lebesgue measure $|\Omega |$. We obtain some properties of the set function $F:\Omega \mapsto {\mathbb{R}}^{+}$ 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 Pólya bound $F(\Omega )\le 1,$ and show that $$F(\Omega )\le 1-{\nu }_{m}T(\Omega )|\Omega {|}^{-1-\frac{2}{m}},$$ where ${\nu }_m$ depends only on $m$. For any $m=2,3,\dots$ and $\epsilon \in (0,1)$ we construct an open set ${\Omega }_{\epsilon }\subset {\mathbb{R}}^m$ such that $F({\Omega }_{\epsilon })\ge 1-\epsilon$.

18 pages, 1 figure