Skip to main content
SpringerLink
Log in

Overdetermined problems for fully nonlinear equations with constant Dirichlet boundary conditions in space forms

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We consider overdetermined problems for two classes of fully nonlinear equations with constant Dirichlet boundary conditions in a bounded domain in space forms. We prove that if the domain is star-shaped, then the solution to the Hessian quotient overdetermined problem is radially symmetric. By establishing a Rellich–Pohožaev type identity for the k-Hessian equation with constant Dirichlet boundary condition, we also show the radial symmetry of the solution to the k-Hessian overdetermined problem for some boundary value without star-shapedness assumption of the domain.

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

References

  1. Bianchini, C., Ciraolo, G.: Wulff shape characterizations in overdetermined anisotropic elliptic problems. Commun. Partial Differ. Equ. 43(5), 790–820 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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  MATH  Google Scholar 

  3. Ciraolo, G., Li, X.: An exterior overdetermined problem for Finsler N-Laplacian in convex cones. Calc. Var. Partial Differ. Equ. 61(4), 121,27 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ciraolo, G., Vezzoni, L.: A rigidity problem on the round sphere. Commun. Contemp. Math. 19(5), 1750001 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ciraolo, G., Vezzoni, L.: On Serrin’s overdetermined problem in space forms. Manuscripta Math. 159(3–4), 445–452 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Farina, A., Roncoroni, A.: Serrin’s type problems in warped product manifolds. Commun. Contemp. Math. 24(4), 2150020 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  8. Garofalo, N., Lewis, J.L.: A symmetry result related to some overdetermined boundary value problems. Amer. J. Math. 111(1), 9–33 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gao, Z., Jia, X., Yan, J.: Serrin-type Overdetermined problems in \(\mathbb{H} ^{n}\). J. Partial Differ. Equ. 36(1), 102–118 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guan, P.: Curvature measures, isoperimetric type inequalities and fully nonlinear PDEs. Fully nonlinear PDEs in real and complex geometry and optics, pp. 47–94, Lecture Notes in Math., 2087, Fond. CIME/CIME Found. Subser., Springer, Cham (2014)

  11. Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988)

    Google Scholar 

  12. Kumaresan, S., Prajapat, J.: Serrin’s result for hyperbolic space and sphere. Duke Math. J. 91(1), 17–28 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lieberman, G.: Second Order Parabolic Differential Equations. World Scientific (1996)

    Book  MATH  Google Scholar 

  14. Ma, X.-N.: A necessary condition of solvability for the capillarity boundary of Monge–Ampere equations in two dimensions. Proc. Am. Math. Soc. 127(3), 763–769 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Molzon, R.: Symmetry and overdetermined boundary value problems. Forum Math. 3, 143–156 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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  MATH  Google Scholar 

  18. Philippin, G.A., Safoui, A.: Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE’s. Z. Angew. Math. Phys. 54(5), 739–755 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Della Pietra, F., Gavitone, N., Xia, C.: Symmetrization with respect to mixed volume. Adv. Math. 388(107887), 1–31 (2021)

    MathSciNet  MATH  Google Scholar 

  20. Poggesi, G.: Radial symmetry for p-harmonic functions in exterior and punctured domains. Appl. Anal. 98(10), 1785–1798 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Qiu, G., Xia, C.: Overdetermined boundary value problems in \(S^{n}\). J. Math. Study 50(2), 165–173 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Reilly, R.C.: On the Hessian of a function and the curvatures of its graph. Michigan. Math. J. 20, 373–383 (1973)

    MathSciNet  MATH  Google Scholar 

  23. Reilly, R.C.: Geometric applications of the solvability of Neumann problems on a Riemannian manifold. Arch. Rational Mech. Anal. 75(1), 23–29 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. Serrin, J.: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43(4), 304–318 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  25. Spruck, J.: Geometric Aspects of the Theory of Fully Nonlinear Elliptic Equations. Global Theory of Minimal Surfaces, Clay Math. Proc., 2, pp. 283–309. Amer. Math. Soc., Providence (2005)

  26. 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  MATH  Google Scholar 

  27. Weinberger, H.F.: Remark on the preceding paper of the Serrin. Arch. Rational Mech. Anal. 43(4), 319–320 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  28. Xia, C., Yin, J.: Two overdetermined problems for anisotropic p-Laplacian. Math. Eng. 4(2), 1–18 (2022)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Chao Qian, Chao Xia and Jiabin Yin for their helpful conversations on this work. We are also grateful to the anonymous reviewer for helpful comments. The first named author is partially supported by Natural Science Basic Research Program of Shaanxi (Program No. 2022JQ-065), The Youth Innovation Team of Shaanxi Universities and Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No. 22JSZ012). The second and the third named authors are partially supported by National Natural Science Foundation of China (Grants No. 11831005 and No. 12061131014). Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, 710119, People’s Republic of China

    Shanze Gao

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

    Hui Ma & Mingxuan Yang

Authors
  1. Shanze Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hui Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mingxuan Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mingxuan Yang.

Additional information

Communicated by Andrea Mondino.

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

Gao, S., Ma, H. & Yang, M. Overdetermined problems for fully nonlinear equations with constant Dirichlet boundary conditions in space forms. Calc. Var. 62, 183 (2023). https://doi.org/10.1007/s00526-023-02533-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-023-02533-3

Mathematics Subject Classification

Search

Navigation