Year: 2016
Type: preprint
Abstract: Let $Ω$ be an open set in Euclidean space with finite Lebesgue measure $|Ω|$. We obtain some properties of the set function $F:Ω\mapsto \R^+$ defined by $$ F(Ω)=\frac{T(Ω)λ_1(Ω)}{|Ω|} ,$$ where $T(Ω)$ and $λ_1(Ω)$ are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical Pólya bound $F(Ω)\le 1,$ and show that $$F(Ω)\le 1- ν_m T(Ω)|Ω|^{-1-\frac2m},$$ where $ν_m$ depends only on $m$. For any $m=2,3,\dots$ and $ε\in (0,1)$ we construct an open set $Ω_ε\subset \R^m$ such that $F(Ω_ε)\ge 1-ε$.
Source: arXiv (Cornell University)
Institutions: University of Bristol, University of Naples Federico II
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Leverhulme Trust
OpenAlex ID: W2284606632