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
Fingerprint
Dive into the research topics of 'On Pólya's inequality for torsional rigidity and first Dirichlet eigenvalue'. Together they form a unique fingerprint.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