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Nearly optimal stability for Serrin’s problem and the Soap Bubble theorem


We present new quantitative estimates for the radially symmetric configuration concerning Serrin’s overdetermined problem for the torsional rigidity, Alexandrov’s Soap Bubble theorem, and other related problems. The new estimates improve on those obtained in Magnanini and Poggesi (J Anal Math, 139(1), 179–205, 2019), Magnanini and Poggesi (Indiana Univ Math J, arXiv:1708.07392, 2017) and are in some cases optimal.

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  1. Aftalion, A., Busca, J., Reichel, W.: Approximate radial symmetry for overdetermined boundary value problems. Adv. Differ. Equ. 4, 907–932 (1999)

    MathSciNet  MATH  Google Scholar 

  2. Alexandrov, A.D.: Uniqueness theorem for surfaces in the large. Vestn. Leningr. Univ. 13(19), 5–8 (1958). Am. Math. Soc. Transl. 21, Ser. 2, 412–416

  3. Alexandrov, A.D.: A characteristic property of spheres. Ann. Math. Pura Appl. 58, 303–315 (1962)

    MathSciNet  Article  Google Scholar 

  4. Brandolini, B., Nitsch, C., Salani, P., Trombetti, C.: On the stability of the Serrin problem. J. Differ. Equ. 245, 1566–1583 (2008)

    MathSciNet  Article  Google Scholar 

  5. Brasco, L., Magnanini, R., Salani, P.: The location of the hot spot in a grounded convex conductor. Indiana Univ. Math. J. 60, 633–660 (2011)

    MathSciNet  Article  Google Scholar 

  6. Boas, H.B., Straube, E.J.: Integral inequalities of Hardy and Poincaré type. Proc. Am. Math. Soc. 103, 172–176 (1998)

    MATH  Google Scholar 

  7. Ciraolo, G., Maggi, F.: On the shape of compact hypersurfaces with almost constant mean curvature. Commun. Pure Appl. Math. 70, 665–716 (2017)

    MathSciNet  Article  Google Scholar 

  8. Ciraolo, G., Magnanini, R., Vespri, V.: Hölder stability for Serrin’s overdetermined problem. Ann. Mat. Pura Appl. 195, 1333–1345 (2016)

    MathSciNet  Article  Google Scholar 

  9. Ciraolo, G., Vezzoni, L.: A sharp quantitative version of Alexandrov’s theorem via the method of moving planes. J. Eur. Math. Soc. 20, 261–299 (2018)

    MathSciNet  Article  Google Scholar 

  10. Feldman, W.M.: Stability of Serrin’s problem and dynamic stability of a model for contact angle motion. SIAM J. Math. Anal. 50–3, 3303–3326 (2018)

    MathSciNet  Article  Google Scholar 

  11. Friedman, A.: Partial Differential Equations. Krieger, Huntington (1983)

    Google Scholar 

  12. Heintze, E., Karcher, H.: A general comparison theorem with applications to volume estimates for submanifolds. Ann. Sci. École Norm. Super. 11, 451–470 (1978)

    MathSciNet  Article  Google Scholar 

  13. Hurri, R.: Poincaré domains in \({\mathbb{R}}^n\). Ann. Acad. Sci. Fenn. Ser. A Math. Dissertationes 71, 1–41 (1988)

  14. Hurri-Syrjänen, R.: An improved Poincaré inequality. Proc. Am. Math. Soc. 120, 213–222 (1994)

    Article  Google Scholar 

  15. Ishiwata, M., Magnanini, R., Wadade, H.: A natural approach to the asymptotic mean value propery for the p-Laplacian. Calc. Var. 56(97), 22 (2017)

    MATH  Google Scholar 

  16. Krummel, B., Maggi, F.: Isoperimetry with upper mean curvature bounds and sharp stability estimates. Calc. Var. Part. Differ. Equ. 56, 53 (2017)

    MathSciNet  Article  Google Scholar 

  17. Magnanini, R.: Alexandrov, Serrin, Weinberger, Reilly: symmetry and stability by integral identities. Bruno Pini Math. Sem. 121–141 (2017)

  18. Magnanini, R., Poggesi, G.: On the stability for Alexandrov’s Soap Bubble theorem. J. Anal. Math. 139(1), 179–205 (2019)

    MathSciNet  Article  Google Scholar 

  19. Magnanini, R., Poggesi, G.: Serrin’s problem and Alexandrov’s Soap Bubble Theorem: enhanced stability via integral identities. Indiana Univ. Math. J. preprint (2017). arXiv:1708.07392

  20. Martio, O., Sarvas, J.: Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Ser. A I Math. 4, 383–401 (1979)

    MathSciNet  Article  Google Scholar 

  21. Payne, L., Schaefer, P.W.: Duality theorems un some overdetermined boundary value problems. Math. Methods Appl. Sci. 11, 805–819 (1989)

    MathSciNet  Article  Google Scholar 

  22. Poggesi, G.: Radial symmetry for \(p\)-harmonic functions in exterior and punctured domains. Appl. Anal. 98, 1785–1798 (2019)

    MathSciNet  Article  Google Scholar 

  23. Poggesi, G.: The Soap Bubble Theorem and Serrin’s problem: quantitative symmetry. Ph.D. thesis, Università di Firenze, defended on February 2019, preprint arXiv:1902.08584

  24. Reilly, R.C.: Applications of the Hessian operator in a Riemannian manifold. Indiana Univ. Math. J. 26, 459–472 (1977)

    MathSciNet  Article  Google Scholar 

  25. Reilly, R.C.: Mean curvature, the Laplacian, and soap bubbles. Am. Math. Mon. 89, 180–188 (1982)

    MathSciNet  Article  Google Scholar 

  26. Serrin, J.: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43, 304–318 (1971)

    MathSciNet  Article  Google Scholar 

  27. Väisälä, J.: Exhaustions of John domains. Ann. Acad. Sci. Fenn. Ser. A I Math. 19, 47–57 (1994)

    MathSciNet  MATH  Google Scholar 

  28. Weinberger, H.F.: Remark on the preceding paper of Serrin. Arch. Ration. Mech. Anal. 43, 319–320 (1971)

    MathSciNet  Article  Google Scholar 

  29. Ziemer, W.P.: A Poincaré type inequality for solutions of elliptic differtential equations. Proc. Am. Math. Soc. 97, 286–290 (1986)

    MATH  Google Scholar 

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The authors wish to thank the anonymous referee, who hinted the estimate (2.21) and whose suggestions contributed to a better presentation of this article. The paper was partially supported by the Gruppo Nazionale Analisi Matematica Probabilità e Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Correspondence to Giorgio Poggesi.

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Communicated by J. Jost.

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Magnanini, R., Poggesi, G. Nearly optimal stability for Serrin’s problem and the Soap Bubble theorem. Calc. Var. 59, 35 (2020).

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Mathematics Subject Classification

  • Primary 35N25
  • 53A10
  • 35B35
  • Secondary 35A23