Optimal Concavity of the Torsion Function

  • Antoine Henrot
  • Carlo Nitsch
  • Paolo Salani
  • Cristina Trombetti


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.


Torsion function Optimal concavity Ellipsoids 

Mathematics Subject Classification

35N25 35R25 35R30 35B06 52A40 


  1. 1.
    Makar-Limanov, L.G.: The solution of the Dirichlet problem for the equation \(\Delta u=-1\) in a convex region, Mat. Zametki 9 (1971) 89–92, English translation. Math. Notes 9, 52–53 (1971) (Russian) Google Scholar
  2. 2.
    Kawohl B.: Rearrangements and convexity of level sets in P.D.E., Lecture Notes in Mathematics, 1150. Springer, Berlin (1985)Google Scholar
  3. 3.
    Kennington, A.U.: Power concavity and boundary value problems. Indiana Univ. Math. J. 34(3), 687–704 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Caffarelli, L.A., Spruck, J.: Convexity of solutions to some classical variational problems. Commun. Partial Differ. Equ. 7, 1337–1379 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Korevaar, N.: Convex solutions to nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math J. 32(4), 603–614 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lindquist P.: A note on the nonlinear Rayleigh quotient. In: Analysis, Algebra, and Computers in Mathematical Research (Lule, 1992), 223231, Lecture Notes in Pure and Appl. Math., 156, Dekker, New York (1994)Google Scholar
  7. 7.
    Salani, P.: A characterization of balls through optimal concavity for potential functions. Proc. AMS 143(1), 173–183 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Serrin, J.: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43, 304–318 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ciraolo, G., Magnanini, R.: A note on Serrin’s overdetermined problem. Kodai Math. J. 37, 728–736 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ciraolo, G., Magnanini, R., Sakaguchi, S.: Symmetry of minimizers with a level surface parallel to the boundary. J. Eur. Math. Soc. 17(11), 2789–2804 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ciraolo, G., Magnanini, R., Vespri, V.: Symmetry and Linear Stability in Serrin’s Overdetermined Problem Via the Stability of the Parallel Surface Problem.
  12. 12.
    Cupini, G., Lanconelli, E.: On an inverse problem in potential theory. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27(4), 431442 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Cupini, G., Lanconelli, E.: Densities with the mean value property for sub-laplacians: an inverse problem, Harmonic analysis, partial differential equations and applications, 109124. Appl. Numer. Harmon. Anal, Birkhuser/Springer, Cham (2017)Google Scholar
  14. 14.
    Magnanini, R., Sakaguchi, S.: Matzoh ball soup: heat conductors with a stationary isothermic surface. Ann. Math. 156, 941–956 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Magnanini, R., Sakaguchi, S.: Nonlinear diffusion with a bounded stationary level surface. Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 937–952 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Schaefer P.W.: On nonstandard overdetermined boundary value problems. In: Proceedings of the Third World Congress of Nonlinear Analysis, Part 4 (Catania, 2000). Nonlinear Analaysis, vol. 47, No. 4, pp. 2203–2212 (2001)Google Scholar
  17. 17.
    Shahgholian, H.: Diversifications of Serrin’s and related problems. Complex Var. Elliptic Equ. 57(6), 653–665 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Brandolini, B., Gavitone, N., Nitsch, C., Trombetti, C.: Characterization of ellipsoids through an overdetermined boundary value problem of Monge–Ampère type. J. Math. Pures Appl. (9) 101(6), 828–841 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Enache, C., Sakaguchi, S.: Some fully nonlinear elliptic boundary value problems with ellipsoidal free boundaries. Math. Nachr. 284(14–15), 1872–1879 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Henrot, A., Philippin, G.A.: Some overdetermined boundary value problems with elliptical free boundaries. SIAM J. Math. Anal. 29(2), 309–320 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

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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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