On relations between principal eigenvalue and torsional rigidity

Michiel van den Berg*, Giuseppe Buttazzo, Aldo Pratelli

*Corresponding author for this work

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


We consider the problem of minimising or maximising the quantity λ(Ω)Tq(Ω) on the class of open sets of prescribed Lebesgue measure. Here q > 0 is fixed, λ(Ω) denotes the first eigenvalue of the Dirichlet Laplacian on H10(Ω), while T(Ω) is the torsional rigidity of Ω. The optimisation problem above is considered in the class of all domains Ω, in the class of convex domains Ω, and in the class of thin domains. The full Blaschke-Santal´o diagram for λ(Ω) and T(Ω) is obtained in dimension one, while for higher dimensions we provide some bounds.
Original languageEnglish
JournalCommunications in Contemporary Mathematics
Publication statusPublished - 17 Dec 2020


  • torsional rigidity
  • shape optimisation
  • principal eigenvalue
  • convex domains

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