JournalsrmiVol. 36, No. 7pp. 2091–2105

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

  • Michiel van den Berg

    University of Bristol, UK
  • Vincenzo Ferone

    Università degli Studi di Napoli Federico II, Italy
  • Carlo Nitsch

    Università degli Studi di Napoli Federico II, Italy
  • Cristina Trombetti

    Università degli Studi di Napoli Federico II, Italy
On a Pólya functional for rhombi, isosceles triangles, and thinning convex sets cover
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Abstract

Let Ω\Omega be an open convex set in Rm\mathbb{R}^m with finite width, and with boundary Ω\partial \Omega. Let vΩv_{\Omega} be the torsion function for Ω\Omega, i.e., the solution of Δv=1,vΩ=0-\Delta v=1, v|_{\partial\Omega}=0. An upper bound is obtained for the product of vΩL(Ω)λ(Ω)\Vert v_{\Omega}\Vert_{L^{\infty}(\Omega)}\lambda(\Omega), where λ(Ω)\lambda(\Omega) is the bottom of the spectrum of the Dirichlet Laplacian acting in L2(Ω)L^2(\Omega). The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area 11, it is shown that vΩL1(Ω)λ(Ω)π2/24\Vert v_{\Omega}\Vert_{L^{1}(\Omega)}\lambda(\Omega)\ge {\pi^2}/{24}, and that this bound is sharp.

Cite this article

Michiel van den Berg, Vincenzo Ferone, Carlo Nitsch, Cristina Trombetti, On a Pólya functional for rhombi, isosceles triangles, and thinning convex sets. Rev. Mat. Iberoam. 36 (2020), no. 7, pp. 2091–2105

DOI 10.4171/RMI/1192