Skip to main content
SpringerLink
Log in

Remarks about the mean value property and some weighted Poincaré-type inequalities

  • Published:
Annali di Matematica Pura ed Applicata (1923 -) Aims and scope Submit manuscript

Abstract

We start providing a quantitative stability theorem for the rigidity of an overdetermined problem involving harmonic functions in a punctured domain. Our approach is inspired by and based on the proof of rigidity established in Enciso and Peralta-Salas (Nonlinear Anal 70(2):1080–1086, 2009), and reveals essential differences with respect to the stability results obtained in the literature for the classical overdetermined Serrin problem. Secondly, we provide new weighted Poincaré-type inequalities for vector fields. These are crucial tools for the study of the quantitative stability issue initiated in Poggesi (Soap bubbles and convex cones: optimal quantitative rigidity, 2022. arXiv:2211.09429) concerning a class of rigidity results involving mixed boundary value problems. Finally, we provide a mean value-type property and an associated weighted Poincaré-type inequality for harmonic functions in cones. A duality relation between this new mean value property and a partially overdetermined boundary value problem is discussed, providing an extension of a classical result obtained in Payne and Schaefer (Math Methods Appl Sci 11(6):805–819, 1989).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. See also [26, 41] for a further recent alternative approach based on a modified implicit function theorem.

  2. More precisely, [10, 20] are based on the proof of rigidity established in [9], whereas [34, 35] are based on the proof in [34, Theorem 2.1] which was inspired by [45, Theorems I.1, I.2]. We refer the interested reader to the surveys [33, 40] for more details.

  3. The star-shapedness of \(\Omega \) is assumed in the case \(N=2\) only to recover the validity of (2.5), but it is not necessary to achieve (possibly different) stability estimates. In fact, as it can be directly checked (from our proof in the case \(N=2\)), such an assumption may be dropped at the cost of using alternative measures for the deviation of \(| \nabla u |\) from being constant on \(\partial \Omega \) and/or allowing the constant C(N) to depend on additional parameters. See also (iii) of Remark 2.3.

  4. In particular, \( \textrm{span}_{x \in \Gamma _1} \nu (x) = \mathbb {R}^N \) is a sufficient condition that guarantees that z must be the origin. See [47] for details.

References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    Google Scholar 

  2. Adolfsson, V., Jerison, D.: \(L^p\)-integrability of the second order derivatives for the Neumann problem in convex domains. Indiana Univ. Math. J. 43, 1123–1138 (1994)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  4. Agostiniani, V., Magnanini, R.: Symmetries in an overdetermined problem for the Green’s function. Discrete Contin. Dyn. Syst. Ser. S4(4), 791–800 (2011)

    MathSciNet  Google Scholar 

  5. Agostiniani, V., Magnanini, R.: Stability in an overdetermined problem for the Green’s function. Ann. Mat. Pura Appl. (4) 190(1), 21–31 (2011)

    Article  MathSciNet  Google Scholar 

  6. Alessandrini, G., Morassi, A., Rosset, E.: The linear constraints in Poincaré and Korn type inequalities. Forum Math. 20, 557–569 (2008)

    Article  MathSciNet  Google Scholar 

  7. Alessandrini, G., Rosset, E.: Symmetry of singular solutions of degenerate quasilinear elliptic equations. Rend. Istit. Mat. Univ. Trieste 39, 1–8 (2007)

    MathSciNet  Google Scholar 

  8. Boas, H.P., Straube, E.J.: Integral inequalities of Hardy and Poincaré type. Proc. Am. Math. Soc. 103(1), 172–176 (1988)

    Google Scholar 

  9. Brandolini, B., Nitsch, C., Salani, P., Trombetti, C.: Serrin-type overdetermined problems: an alternative proof. Arch. Ration. Mech. Anal. 190(2), 267–280 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Cabrè, X., Ros-Oton, X., Serra, J.: Sharp isoperimetric inequalities via the ABP method. J. Eur. Math. Soc. (JEMS) 18(12), 2971–2998 (2016)

    Article  MathSciNet  Google Scholar 

  12. Choe, J., Park, S.-H.: Capillary surfaces in a convex cone. Math. Z. 267, 875–886 (2011)

    Article  MathSciNet  Google Scholar 

  13. Cianchi, A., Salani, P.: Overdetermined anisotropic elliptic problems. Math. Ann. 345(4), 859–881 (2009)

    Article  MathSciNet  Google Scholar 

  14. Ciraolo, G., Figalli, A., Roncoroni, A.: Symmetry results for critical anisotropic p-Laplacian equations in convex cones. Geom. Funct. Anal. 30(3), 770–803 (2020)

    Article  MathSciNet  Google Scholar 

  15. Ciraolo, G., Li, X.: An exterior overdetermined problem for Finsler N-Laplacian in convex cones. Calc. Var. Partial Differ. Equ. 61(4), Paper No. 121 (2022)

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

    Article  MathSciNet  Google Scholar 

  17. Cozzi, M., Farina, A., Valdinoci, E.: Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs. Adv. Math. 293, 343–381 (2016)

    Article  MathSciNet  Google Scholar 

  18. Dipierro, S., Poggesi, G., Valdinoci, E.: Radial symmetry of solutions to anisotropic and weighted diffusion equations with discontinuous nonlinearities. Calc. Var. Partial Differ. Equ. 61(2), Paper No. 72 (2022)

  19. Enciso, A., Peralta-Salas, D.: Daniel Symmetry for an overdetermined boundary problem in a punctured domain. Nonlinear Anal. 70(2), 1080–1086 (2009)

    Article  MathSciNet  Google Scholar 

  20. 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)

    Article  MathSciNet  Google Scholar 

  21. Ferone, V., Kawohl, B.: Remarks on a Finsler–Laplacian. Proc. Am. Math. Soc. 137(1), 247–253 (2009)

    Article  MathSciNet  Google Scholar 

  22. Figalli, A., Indrei, E.: A sharp stability result for the relative isoperimetric inequality inside convex cones. J. Geom. Anal. 23(2), 938–969 (2013)

    Article  MathSciNet  Google Scholar 

  23. Figalli, A., Maggi, F., Pratelli, A.: A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182(1), 167–211 (2010)

    Article  MathSciNet  Google Scholar 

  24. Fusco, N., Julin, V.: A strong form of the quantitative isoperimetric inequality. Calc. Var. Partial Differ. Equ. 50(3–4), 925–937 (2014)

    Article  MathSciNet  Google Scholar 

  25. Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality. Ann. Math. (2) 168(3), 9410–980 (2008)

    Article  MathSciNet  Google Scholar 

  26. Gilsbach, A., Onodera, M.: Linear stability estimates for Serrin’s problem via a modified implicit function theorem. Calc. Var. Partial Differ. Equ. 60(6), Paper No. 241 (2021)

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

    MathSciNet  Google Scholar 

  28. Indrei, E.: A weighted relative isoperimetric inequality in convex cones. Methods Appl. Anal. 28(1), 1–13 (2021)

    Article  MathSciNet  Google Scholar 

  29. Ishiwata, M., Magnanini, R., Wadade, H.: A natural approach to the asymptotic mean value property for the p-Laplacian. Calc. Var. Partial Differ. Equ. 56(4), Paper No. 97 (2017)

  30. Kichenassamy, S., Véron, L.: Singular solutions of the p-Laplace equation. Math. Ann. 275, 599–615 (1986)

    Article  MathSciNet  Google Scholar 

  31. Kufner, A.: Weighted Sobolev Spaces. Wiley, New York (1985)

    Google Scholar 

  32. Lions, P.L., Pacella, F.: Isoperimetric inequalities for convex cones. Proc. Am. Math. Soc. 109, 477–485 (1990)

    Article  MathSciNet  Google Scholar 

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

  34. Magnanini, R., Poggesi, G.: Serrin’s problem and Alexandrov’s Soap Bubble theorem: stability via integral identities. Indiana Univ. Math. J. 69(4), 1181–1205 (2020)

    Article  MathSciNet  Google Scholar 

  35. Magnanini, R., Poggesi, G.: Nearly optimal stability for Serrin’s problem and the Soap Bubble theorem. Calc. Var. Partial Differ. Equ. 59(1), Paper No. 35 (2020)

  36. Magnanini, R., Poggesi, G.: Interpolating estimates with applications to some quantitative symmetry results. Math. Eng. 5(1), Paper No. 002 (2023)

  37. Maz’ya, V.: On the boundedness of first derivatives for solutions to the Neumann–Laplace problem in a convex domain. J. Math. Sci. (N.Y.) 159, 104–112 (2009)

    Article  MathSciNet  Google Scholar 

  38. Meyers, N.G.: Integral inequalities of Poincaré and Wirtinger type. Arch. Ration. Mech. Anal. 68(2), 113–120 (1978)

    Article  Google Scholar 

  39. Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer, New York (2009)

    Google Scholar 

  40. Nitsch, C., Trombetti, C.: The classical overdetermined Serrin problem Complex Var. Elliptic Equ. 63(7–8), 1107–1122 (2018)

    MathSciNet  Google Scholar 

  41. Onodera, M.: Linear stability analysis of overdetermined problems with non-constant data. Math. Eng. 5(3), Paper No. 048 (2023)

  42. Pacella, F., Poggesi, G., Roncoroni, A.: Optimal quantitative stability for a Serrin-type problem in convex cones (forthcoming)

  43. Pacella, F., Tralli, G.: Overdetermined problems and constant mean curvature surfaces in cones. Rev. Mat. Iberoam. 36(3), 841–867 (2020)

    Article  MathSciNet  Google Scholar 

  44. Pacella, F., Tralli, G.: Isoperimetric cones and minimal solutions of partial overdetermined problems. Publ. Mat. 65(1), 61–81 (2021)

    Article  MathSciNet  Google Scholar 

  45. Payne, L.E., Schaefer, P.W.: Duality theorems in some overdetermined boundary value problems. Math. Methods Appl. Sci. 11(6), 805–819 (1989)

    Article  MathSciNet  Google Scholar 

  46. Philippin, G.A.: On a free boundary problem in Electrostatics. Math. Methods Appl. Sci. 12, 387–392 (1990)

    Article  MathSciNet  Google Scholar 

  47. Poggesi, G.: Soap bubbles and convex cones: optimal quantitative rigidity. Preprint (2022). arXiv:2211.09429

  48. Ritoré, M., Rosales, C.: Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones. Trans. Am. Math. Soc. 356(11), 4601–4622 (2004)

    Article  MathSciNet  Google Scholar 

  49. Scheuer, J.: Stability from rigidity via umbilicity. Preprint (2021). arXiv:2103.07178

  50. Serrin, J.: Isolated singularities of solutions of quasi-linear equations. Acta Math. 113, 219–240 (1965)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  53. Wang, G., Xia, C.: A characterization of the Wulff shape by an overdetermined anisotropic PDE. Arch. Ration. Mech. Anal. 199(1), 99–115 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  55. Weng, L.: An overdetermined problem of anisotropic equations in convex cones. J. Differ. Equ. 268(7), 3646–3664 (2020)

    Article  MathSciNet  Google Scholar 

  56. Xia, C., Yin, J.: Two overdetermined problems for anisotropic p-Laplacian. Math. Eng. 4(2), Paper No. 015 (2022)

  57. Ziemer, W.P.: A Poincaré-type inequality for solutions of elliptic differential equations. Proc. Am. Math. Soc. 97(2), 286–290 (1986)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is supported by the Australian Research Council (ARC) Discovery Early Career Researcher Award (DECRA) DE230100954 “Partial Differential Equations: geometric aspects and applications” and the 2023 J G Russell Award from the Australian Academy of Science. The author is member of the Australian Mathematical Society (AustMS) and the Gruppo Nazionale Analisi Matematica Probabilità e Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The author thanks Rolando Magnanini for bringing to his attention [4, 5] and the content of [17, Theorem 1.2].

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA, 6009, Australia

    Giorgio Poggesi

Authors
  1. Giorgio Poggesi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giorgio Poggesi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Poggesi, G. Remarks about the mean value property and some weighted Poincaré-type inequalities. Annali di Matematica (2023). https://doi.org/10.1007/s10231-023-01408-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10231-023-01408-w

Keywords

Mathematics Subject Classification

Search

Navigation