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

• Michiel van den Berg ^[1] ; Vincenzo Ferone ^[2] ; Carlo Nitsch ^[2] ; Cristina Trombetti ^[2]
  1. [1] University of Bristol

     University of Bristol

     Reino Unido

     
  2. [2] University of Naples Federico II

     University of Naples Federico II

     Nápoles, Italia

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 7, 2020, págs. 2091-2105
• Idioma: inglés
• Enlaces
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• Resumen

  • Let Ω be an open convex set in Rm with finite width, and with boundary ∂Ω. Let vΩ be the torsion function for Ω, i.e., the solution of −Δv=1,v|∂Ω=0. 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)≥π2/24, and that this bound is sharp.