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arXiv · 1602.04618

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

M. van den Berg·V. Ferone·C. Nitsch·C. Trombetti
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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 $F(\Omega_{\epsilon})\ge 1-\epsilon$.

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M. van den Berg, V. Ferone, C. Nitsch, C. Trombetti. 2016-02-15. On Polya's inequality for torsional rigidity and first Dirichlet eigenvalue. https://arxiv.org/abs/1602.04618

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