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

Michiel van den Berg, Vincenzo Ferone, Carlo Nitsch, Cristina Trombetti

Research output: Contribution to journalArticle (Academic Journal)peer-review

29 Citations (Scopus)
253 Downloads (Pure)


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 Fϵ)≥1−ϵ.
Original languageEnglish
Pages (from-to)579-600
Number of pages23
JournalIntegral Equations and Operator Theory
Issue number4
Early online date9 Nov 2016
Publication statusPublished - Dec 2016


  • Torsional rigidity
  • first Dirichlet eigenvalue


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