Skip to main content
SpringerLink
Log in

Rigidity and quantitative stability for partially overdetermined problems and capillary CMC hypersurfaces

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

Abstract

In this paper, we first prove a rigidity result for a Serrin-type partially overdetermined problem in the half-space, which gives a characterization of capillary spherical caps by the overdetermined problem. In the second part, we prove quantitative stability results for the Serrin-type partially overdetermined problem, as well as capillary almost constant mean curvature hypersurfaces in the half-space.

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

Data availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Aleksandrov, A.D.: Uniqueness theorems for surfaces in the large. Am. Math. Soc. 21(341–354), 354–388 (1962). https://doi.org/10.1090/trans2/021/09

    Article  MathSciNet  Google Scholar 

  2. Ainouz, A., Souam, R.: Stable capillary hypersurfaces in a half-space or a slab. Indiana Univ. Math. J. 65(3), 813–831 (2016). https://doi.org/10.1512/iumj.2016.65.5839

    Article  MathSciNet  Google Scholar 

  3. Brandolini, B., Nitsch, C., Salani, P., Trombetti, C.: On the stability of the Serrin problem. J. Differ. Equ. 245(6), 1566–1583 (2008). https://doi.org/10.1016/j.jde.2008.06.010

    Article  MathSciNet  Google Scholar 

  4. Ciraolo, G., Li, X.: An exterior overdetermined problem for Finsler \(N\)-Laplacian in convex cones. Calc. Var. Part. Differ. Equ. 61(4), 121 (2022). https://doi.org/10.1007/s00526-022-02235-2

    Article  MathSciNet  Google Scholar 

  5. Ciraolo, G., Maggi, F.: On the shape of compact hypersurfaces with almost-constant mean curvature. Comm. Pure Appl. Math. 70(4), 665–716 (2017). https://doi.org/10.1002/cpa.21683

    Article  MathSciNet  Google Scholar 

  6. De Lellis, C., Müller, S.: Optimal rigidity estimates for nearly umbilical surfaces. J. Differ. Geom. 69(1), 75–110 (2005). https://doi.org/10.4310/jdg/1121540340

    Article  MathSciNet  Google Scholar 

  7. De Lellis, C., Müller, S.: A \(C^0\) estimate for nearly umbilical surfaces. Calc. Var. Partial. Differ. Equ. 26(3), 283–296 (2006). https://doi.org/10.1007/s00526-006-0005-5

    Article  Google Scholar 

  8. Delgadino, M.G., Maggi, F., Mihaila, C., Neumayer, R.: Bubbling with \(L^2\)-almost constant mean curvature and an Alexandrov-type theorem for crystals. Arch. Ration. Mech. Anal. 230(3), 1131–1177 (2018). https://doi.org/10.1007/s00205-018-1267-8

    Article  MathSciNet  Google Scholar 

  9. De Rosa, A., Gioffrè, S.: Quantitative stability for anisotropic nearly umbilical hypersurfaces. J. Geom. Anal. 29(3), 2318–2346 (2019). https://doi.org/10.1007/s12220-018-0079-2

    Article  MathSciNet  Google Scholar 

  10. De Rosa, A., Gioffrè, S.: Absence of bubbling phenomena for non-convex anisotropic nearly umbilical and quasi-Einstein hypersurfaces. J. Reine Angew. Math. 780, 1–40 (2021). https://doi.org/10.1515/crelle-2021-0038

    Article  MathSciNet  Google Scholar 

  11. Finn, R.: Equilibrium Capillary Surfaces, vol. 284. Springer, Cham (1986)

    Google Scholar 

  12. Ghomi, M.: Gauss map, topology, and convexity of hypersurfaces with nonvanishing curvature. Topology 41(1), 107–117 (2002). https://doi.org/10.1016/S0040-9383(00)00028-8

    Article  MathSciNet  Google Scholar 

  13. Guo, J., Xia, C.: A partially overdetermined problem in a half ball. Calc. Var. Part. Differ. Equ. 58(5), 15 (2019). https://doi.org/10.1007/s00526-019-1603-3

    Article  MathSciNet  Google Scholar 

  14. Jia, X., Wang, G., Xia, C., Zhang, X.: Heintze-Karcher inequality and capillary hypersurfaces in a wedge. In: Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) (2022). arXiv:2209.13839, https://doi.org/10.2422/2036-2145.202212_001

  15. Jia, X., Wang, G., Xia, C., Zhang, X.: Alexandrov’s theorem for anisotropic capillary hypersurfaces in the half-space. Arch. Ration. Mech. Anal. 247(2), 25 (2023). https://doi.org/10.1007/s00205-023-01861-0

    Article  MathSciNet  Google Scholar 

  16. Jia, X., Xia, C., Zhang, X.: A Heintze–Karcher-type inequality for hypersurfaces with capillary boundary. J. Geom. Anal. 33(6), 19 (2023). https://doi.org/10.1007/s12220-023-01230-z

    Article  MathSciNet  Google Scholar 

  17. Jia, X., Zhang, X.: Quantitative Alexandrov theorem for capillary hypersurfaces in the half-space (2024). arXiv:2403.06597

  18. Liebermann, G.M.: Mixed boundary value problems for elliptic and parabolic differential equations of second order. J. Math. Anal. Appl. 113, 422–440 (1986). https://doi.org/10.1016/0022-247X(86)90314-8

    Article  MathSciNet  Google Scholar 

  19. Liebermann, G.M.: Optimal Hölder regularity for mixed boundary value problems. J. Math. Anal. Appl. 143(2), 572–586 (1989). https://doi.org/10.1016/0022-247X(89)90061-9

    Article  MathSciNet  Google Scholar 

  20. Magnanini, R., Poggesi, G.: On the stability for Alexandrov’s soap bubble theorem. J. Anal. Math. 139(1), 179–205 (2019). https://doi.org/10.1007/s11854-019-0058-y

    Article  MathSciNet  Google Scholar 

  21. Magnanini, R., Poggesi, G.: Nearly optimal stability for Serrin’s problem and the soap bubble theorem. Calc. Var. Part. Differ. Equ. 59(1), 23 (2020). https://doi.org/10.1007/s00526-019-1689-7

    Article  MathSciNet  Google Scholar 

  22. 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). https://doi.org/10.1512/iumj.2020.69.7925

    Article  MathSciNet  Google Scholar 

  23. Magnanini, R., Poggesi, G.: Quantitative symmetry in a mixed Serrin-type problem for a constrained torsional rigidity. Calc. Var. Part. Diff. Equ. 63(1), 26 (2024). https://doi.org/10.1007/s00526-023-02629-w

    Article  MathSciNet  Google Scholar 

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

  25. Reilly, R.C.: Applications of the Hessian operator in a Riemannian manifold. Indiana Univ. Math. J. 26, 459–472 (1977). https://doi.org/10.1512/iumj.1977.26.26036

    Article  MathSciNet  Google Scholar 

  26. Ros, A.: Compact hypersurfaces with constant higher order mean curvatures. Rev. Mat. Iberoam. 3(3–4), 447–453 (1987). https://doi.org/10.4171/RMI/58

    Article  MathSciNet  Google Scholar 

  27. Scheuer, J.: Stability from rigidity via umbilicity. In: Adv. Calc. Var. (2021). arXiv: 2103.07178

  28. Serrin, J.: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43, 304–318 (1971). https://doi.org/10.1007/BF00250468

    Article  MathSciNet  Google Scholar 

  29. Scheuer, J., Xia, C.: Stability for Serrin’s problem and Alexandroff’s theorem in warped product manifolds. Int. Math. Res. Not. IMRN 24, 21086–21108 (2023). https://doi.org/10.1093/imrn/rnac294

    Article  MathSciNet  Google Scholar 

  30. Scheuer, J., Zhang, X.: Stability of the Wulff shape with respect to anisotropic curvature functionals (2023). arXiv: 2308.15999

  31. Weinberger, H.F.: Remark on the preceding paper of Serrin. Arch. Ration. Mech. Anal. 43, 319–320 (1971). https://doi.org/10.1007/BF00250469

    Article  MathSciNet  Google Scholar 

  32. Wente, H.: The symmetry of sessile and pendent drops. Pac. J. Math. 88, 387–397 (1980). https://doi.org/10.2140/pjm.1980.88.387

    Article  MathSciNet  Google Scholar 

  33. Wang, G., Weng, L., Xia, C.: Alexandrov–Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary. Math. Ann. 388(2), 2121–2154 (2024). https://doi.org/10.1007/s00208-023-02571-4

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the referee for careful reading and valuable suggestions which helped improve the paper.

Author information

Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 211189, People’s Republic of China

    Xiaohan Jia

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China

    Zheng Lu, Chao Xia & Xuwen Zhang

  3. Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, 79104, Freiburg im Breisgau, Germany

    Zheng Lu

  4. Institut für Mathematik, Goethe-Universität, 60325, Frankfurt, Germany

    Xuwen Zhang

Authors
  1. Xiaohan Jia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zheng Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chao Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xuwen Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chao Xia.

Additional information

Communicated by Y. Tonegawa.

Publisher's Note

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

This work is supported by the NSFC (Grant Nos. 12271449, 12126102).

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

Jia, X., Lu, Z., Xia, C. et al. Rigidity and quantitative stability for partially overdetermined problems and capillary CMC hypersurfaces. Calc. Var. 63, 125 (2024). https://doi.org/10.1007/s00526-024-02733-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-024-02733-5

Mathematics Subject Classification

Search

Navigation