Skip to main content
Log in

Hölder regularity for equations of prescribed anisotropic mean curvature type

  • Published:
Manuscripta Mathematica Aims and scope Submit manuscript

Abstract

In this note, we prove Hölder regularity for equations of prescribed anisotropic mean curvature type. As an application, we obtain the regularity of weak surfaces with prescribed anisotropic mean curvature.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Almgren F.J. Jr.: Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Ann. Math. (2) 87, 321–391 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bethuel F.: Un rsultat de rgularit pour les solutions de l’quation de surfaces ă©1 courbure moyenne prescrite. (French) [A regularity result for solutions to the equation of surfaces of prescribed mean curvature]. C. R. Acad. Sci. Paris Sr. I Math. 314(13), 1003–1007 (1992)

    MathSciNet  MATH  Google Scholar 

  3. Coifman R., Lions P.L., Meyer Y., Semmes S.: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 72, 247–286 (1993)

    MathSciNet  MATH  Google Scholar 

  4. Federer H.: Geometric Measure Theory. Springer, New York (1969)

    MATH  Google Scholar 

  5. Fefferman C., Stein E.: H p space of several variables. Acta Math. 129, 137–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  6. Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies, 105. Princeton University Press, Princeton, NJ (1983)

  7. Grüter M.: Regularity of weak H-surfaces. J. Reine Angew. Math. 329, 1–15 (1981)

    MathSciNet  MATH  Google Scholar 

  8. He Y.J., Li H.Z., Ma H., Ge J.Q.: Compact embedded hypersurfaces with constant higher order anisotropic mean curvature. Indiana Univ. Math. J. 58, 853–868 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hildebrandt S., Kaul H.: Two-dimensional variational problems with obstructions, and Plateau’s problem for H-surfaces in a Riemannian manifold. Commun. Pure Appl. Math. 25, 187–223 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hildebrandt S., von der Mosel H.: Plateau’s problem for parametric double integrals. I. Existence and regularity in the interior. Commun. Pure Appl. Math. 56(7), 926–955 (2003)

    Article  MATH  Google Scholar 

  11. Hildebrandt S., von der Mosel H.: Plateau’s problem for parametric double integrals. II. Regularity at the boundary. J. Reine Angew. Math. 565, 207–233 (2003)

    MathSciNet  MATH  Google Scholar 

  12. Hildebrandt S., von der Mosel H.: Conformal representation of surfaces, and Plateau’s problem for Cartan functionals. Riv. Mat. Univ. Parma (7) 4, 1–43 (2005)

    Google Scholar 

  13. Iwaniec T.: p-harmonic tensors and quasiregular mappings. Ann. Math. 136(3), 589–624 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. John F., Nirenberg L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  15. Koiso M., Palmer B.: Anisotropic umbilic points and Hopf’s theorem for surfaces with constant anisotropic mean curvature. Indiana Univ. Math. J. 59(1), 79–90 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Palmer B.: Stability of the Wulff shape. Proc. Am. Math. Soc. 126(12), 3661–3667 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Riviere T.: Conservation laws for conformally invariant variational problems. Invent. Math. 168(1), 1–22 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Strzelecki P.: Regularity of p-harmonic maps from the p-dimensional ball into a sphere. Manuscr. Math. 82(3–4), 407–415 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Strzelecki P.: A new proof of regularity of weak solutions of the H-surface equation. Calc. Var. Partial Differ. Equ. 16(3), 227–242 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wente H.C.: An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl. 26, 318–344 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhai J.: Regularity of weak constant anisotropic mean curvature surfaces. C. R. Math. Acad. Sci. Paris 344(9), 603–606 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Chao Xia.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Luo, Y., Xia, C. Hölder regularity for equations of prescribed anisotropic mean curvature type. manuscripta math. 141, 589–600 (2013). https://doi.org/10.1007/s00229-012-0584-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-012-0584-8

Mathematics subject classification (2010)

Navigation