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→R^{+} defined by
F(Ω)=T(Ω)λ1(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)≤1, and show that F(Ω)≤1−ν_{m}T(Ω)Ω^{−1−2/m}, where ν_{m} depends on m only. For any m=2,3,… and ϵ ∈ (0,1) we construct an open set Ω_{ϵ }⊂ R^{m} such that F(Ω_{ϵ})≥1−ϵ.
F(Ω)=T(Ω)λ1(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)≤1, and show that F(Ω)≤1−ν_{m}T(Ω)Ω^{−1−2/m}, where ν_{m} depends on m only. For any m=2,3,… and ϵ ∈ (0,1) we construct an open set Ω_{ϵ }⊂ R^{m} such that F(Ω_{ϵ})≥1−ϵ.
Original language  English 

Pages (fromto)  579600 
Number of pages  23 
Journal  Integral Equations and Operator Theory 
Volume  86 
Issue number  4 
Early online date  9 Nov 2016 
DOIs  
Publication status  Published  Dec 2016 
Keywords
 Torsional rigidity
 first Dirichlet eigenvalue
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Profiles

Professor Michiel Van Den Berg
 School of Mathematics  Emeritus Professor
 Probability, Analysis and Dynamics
 Pure Mathematics
 Analysis
Person: Academic , Member, Honorary and Visiting Academic