publication . Preprint . Article . 2016

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

van den Berg, Michiel; Ferone, Vincenzo; Nitsch, Carlo; Trombetti, Cristina;
Open Access English
  • Published: 09 Nov 2016
  • Country: United Kingdom
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-\e...
Subjects
arXiv: Mathematics::Spectral Theory
free text keywords: Torsional rigidity, first Dirichlet eigenvalue, Mathematics - Analysis of PDEs, 49J45, 49R05, 35P15, 47A75, 35J25, Algebra and Number Theory, Analysis, Euclidean space, Lambda, Open set, Lebesgue measure, Mathematical analysis, Set function, Dirichlet eigenvalue, Omega, Combinatorics, Eigenvalues and eigenvectors, Mathematics
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