Skip to main content
SpringerLink
Log in

Partial regularity for Lipschitz solutions to the minimal surface system

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

Abstract

In this paper, we study the regularity of several notions of Lipschitz solutions to the minimal surface system with an emphasis on partial regularity results. These include stationary solutions, integral weak solutions, and viscosity solutions. We also prove an interior gradient estimate for classical solutions to the system using the maximum principle, assuming the area-decreasing condition and that all but one component have small \(L^\infty \) norm.

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.

Notes

  1. That is, F is a bi-Lipschitz map with \(\Gamma = F(\Omega )\).

  2. See Definition 2.3 below and the original work [18].

  3. In the sense of Hausdorff dimension.

  4. Here, \(A \triangle B\) denotes the symmetric difference \((A\setminus B) \cup (B \setminus A)\).

  5. That is, if the graph of a viscosity solution \(u:B_1^n(0) \rightarrow {\mathbb {R}}^m\) touches a hyperplane \(L \subset {\mathbb {R}}^{n+m}\) from one side it lies entirely in L.

  6. As before, \(\Delta _L h\) denotes the Laplacian of h restricted to the n-plane L.

References

  1. Allard, W.K.: On the first variation of a varifold. Ann. Math. 95(3), 417–491 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  2. Armstrong, S.N., Silvestre, L., Smart, C.K.: Partial regularity of solutions of fully nonlinear, uniformly elliptic equations. Comm. Pure Appl. Math 65(8), 1169–1184 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence, RI, (1995)

  4. Douglas, J.: Solution of the problem of Plateau. Trans. Amer. Math. Soc. 33(1), 263–321 (1931)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fischer-Colbrie, D.: Some rigidity theorems for minimal submanifolds of the sphere. Acta Math. 145(1–2), 29–46 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  6. Giusti, E.: Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80. Birkhäuser Verlag, Basel (1984)

    Book  Google Scholar 

  7. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Reprint of 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, (2001)

  8. Jensen, R.: Uniformly elliptic PDEs with bounded, measurable coefficients. J Fourier Anal Appl. 2(3), 237–259 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jost, J., Xin, Y.L.: Berstein type theorems for higher codimension. Calc. Var. Partial Differ. Equ. 9(4), 277–296 (1999)

    Article  MATH  Google Scholar 

  10. Korevaar, N.: An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation. Nonlinear Functional Analysis and its Applications, Part 2 (Berkeley, Calif., 1983), 81-89, Proc. Sympos. Pure Math., 45, Part 2, Amer. Math. Soc., Providence, RI, (1986)

  11. Lawson, H.B., Osserman, R.: Non-existence, non-uniqueness, and irregularity of solutions to the minimal surface system. Acta Math. 139(1–2), 1–17 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  12. Martinazzi, L., Giaquinta, M.: An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs. Second edition. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), 11. Edizioni della Normale, Pisa, (2012)

  13. Mooney, C.: A proof of the Krylov-Safonov theorem without localization. Comm. Partial Differ. Equ. 44(8), 681–690 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nadirashvili, N., Vladut, S.: Homogeneous solutions of fully nonlinear elliptic equations in four dimensions. Comm. Pure Appl. Math. 66(10), 1653–1662 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nadirashvili, N., Vladut, S.: Singular solution to special Lagrangian equations. Ann. Inst. H. Poincaré C Anal. Non Linéaire 27(5), 1179–1188 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Rado, T.: On Plateau’s problem. Ann. Math. 31(3), 457–469 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  17. Savin, O.: Small perturbation solutions for elliptic equations. Comm. Partial Differ. Equ. 32(4–6), 557–578 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Savin, O.: Viscosity solutions and the minimal surface system. Nonlinear analysis in geometry and applied mathematics. Part 2, 135–145, Harv. Univ. Cent. Math. Sci. Appl. Ser. Math., 2, Int. Press, Somerville, MA, (2018)

  19. Simon, L.: Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, 3. Australian National University, Centre for Mathematical Analysis, Canberra, (1983)

  20. Wang, M.T.: Interior gradient bounds for solutions to the minimal surface system. Am. J. Math. 126(4), 921–934 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang, M.T.: The Dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions. Comm. Pure Appl. Math. 57(2), 267–281 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang, D., Yuan, Y.: Singular solutions to special Lagrangian equations with subcritical phases and minimal surface systems. Amer. J. Math. 135(5), 1157–1177 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I gratefully acknowledge the support as a graduate student researcher through C. Mooney’s Sloan Fellowship and UC Irvine Chancellor’s Fellowship. I would also like to thank my thesis advisers C. Mooney and R. Schoen for their patient guidance and feedback on the drafts of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, UC Irvine, Irvine, California, USA

    Bryan Dimler

Authors
  1. Bryan Dimler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bryan Dimler.

Additional information

Communicated by Laszlo Szekelyhidi.

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

Dimler, B. Partial regularity for Lipschitz solutions to the minimal surface system. Calc. Var. 62, 260 (2023). https://doi.org/10.1007/s00526-023-02591-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-023-02591-7

Mathematics Subject Classification

Search

Navigation