On a Pólya functional for rhombi, isosceles triangles, and thinning convex sets

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

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Abstract

Let Ω be an open convex set in Rm with finite width, and let vΩ be the torsion function for Ω, i.e. the solution of −Δv=1,v∈H10(Ω). An upper bound is obtained for the product of ∥vΩ∥L∞(Ω)λ(Ω), where λ(Ω) is the bottom of the spectrum of the Dirichlet Laplacian acting in L2(Ω). The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area 1, it is shown that ∥vΩ∥L1(Ω)λ(Ω)≥π224, and that this bound is sharp
Original languageEnglish
Pages (from-to)2091-2105
Number of pages15
JournalRevista Matemática Iberoamericana
Volume36
Issue number7
Early online date16 Mar 2020
DOIs
Publication statusE-pub ahead of print - 16 Mar 2020

Keywords

  • torsion function
  • torsional rigidity
  • first Dirichlet eigenvalue

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