Optimal Concavity of the Torsion Function

  • Antoine Henrot
  • Carlo Nitsch
  • Paolo Salani
  • Cristina Trombetti
Article
  • 21 Downloads

Abstract

It is well known that the torsion function of a convex domain has a square root which is concave. The power one half is optimal in the sense that no greater power ensures concavity for every convex set. In this paper, we investigate concavity, not of a power of the torsion function itself, but of the complement to its maximum. Requiring that the torsion function enjoys such a property for the power one half leads to an unconventional overdetermined problem. Our main result is to show that solutions of this problem exist, if and only if they are quadratic polynomials, finding, in fact, a new characterization of ellipsoids.

Keywords

Torsion function Optimal concavity Ellipsoids 

Mathematics Subject Classification

35N25 35R25 35R30 35B06 52A40 

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Copyright information

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Authors and Affiliations

  1. 1.Institut Élie Cartan de Lorraine UMR7502Université de Lorraine - CNRSNancyFrance
  2. 2.Dipartimento di Matematica e Applicazioni R. CaccioppoliUniversità degli Studi di Napoli “Federico II”NaplesItaly
  3. 3.Dipartimento di Matematica e Informatica “U. Dini”Università degli Studi di FirenzeFlorenceItaly

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