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 λ1(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−νmT(Ω)|Ω|−1−2/m, where νm depends on m only. For any m=2,3,… and ϵ ∈ (0,1) we construct an open set Ωϵ ⊂ Rm such that F(Ωϵ)≥1−ϵ.
F(Ω)=T(Ω)λ1(Ω)/|Ω|, where T(Ω) and λ1(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−νmT(Ω)|Ω|−1−2/m, where νm depends on m only. For any m=2,3,… and ϵ ∈ (0,1) we construct an open set Ωϵ ⊂ Rm such that F(Ωϵ)≥1−ϵ.
| Original language | English |
|---|---|
| Pages (from-to) | 579-600 |
| 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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Professor Michiel van den Berg
- School of Mathematics - Emeritus Professor
- Probability, Analysis and Dynamics
- Pure Mathematics
- Analysis
Person: Member, Honorary and Visiting Academic