Skip to main content
SpringerLink
Log in

Power Convexity of Solutions to a Special Lagrangian Equation in Dimension Two

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

In this paper, we prove power convexity result of solution to Dirichlet problem of special Lagrangian equation in dimension two. This provides new example of fully nonlinear elliptic boundary value problem whose solution shares power convexity property previously only knew for 2-Hessian equation in dimension three. The key ingredients consist of microscopic convexity principles and deformation methods.

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. Acker, A., Payne, L.E., Philippin, G.: On the convexity of level lines of the fundamental mode in the clamped membrane problem, and the existence of convex solutions in a related free boundary problem. Z. Angew. Math. Phys. 32(6), 683–694 (1981)

    MathSciNet  MATH  Google Scholar 

  2. Ahlfors, L.V.: Conformal Invariants. Topics in Geometric Function Theory Reprint of the 1973 Original, AMS Chelsea Publishing, Providence (2010)

    MATH  Google Scholar 

  3. Alvarez, O., Lasry, J.-M., Lions, P.-L.: Convex viscosity solutions and state constraints. J. Math. Pures Appl. 76(3), 265–288 (1997)

    MathSciNet  MATH  Google Scholar 

  4. Bhattacharya, A.: The Dirichlet problem for Lagrangian mean curvature equation. arXiv:2005.14420 (2020)

  5. Bian, B., Guan, P.: A microscopic convexity principle for nonlinear partial differential equations. Invent. Math. 177(2), 307–335 (2009)

    MathSciNet  MATH  Google Scholar 

  6. Bianchini, M., Salani, P.: Power concavity for solutions of nonlinear elliptic problems in convex domains. In: Geometric Properties for Parabolic and Elliptic PDE’s, 35–48, Springer INdAM Ser. 2, Springer, Milan (2013)

  7. Bianchini, C., Longinetti, M., Salani, P.: Quasiconcave solutions to elliptic problems in convex rings. Indiana Univ. Math. J. 58(4), 1565–1589 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Borrelli, W., Mosconi, S., Squassina, M.: Concavity properties for solutions to \(p\)-Laplace equations with concave nonlinearities. arXiv:2111.14801 (2021)

  9. Brascamp, H.J., Lieb, E.H.: On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22(4), 366–389 (1976)

    MATH  Google Scholar 

  10. Caffarelli, L.A., Friedman, A.: Convexity of solutions of semilinear elliptic equations. Duke Math. J. 52(2), 431–456 (1985)

    MathSciNet  MATH  Google Scholar 

  11. Caffarelli, L.A., Spruck, J.: Convexity properties of solutions to some classical variational problems. Commun. Partial Diff. Equ. 7(11), 1337–1379 (1982)

    MathSciNet  MATH  Google Scholar 

  12. Caffarelli, L.A., Guan, P., Ma, X.-N.: A constant rank theorem for solutions of fully nonlinear elliptic equations. Commun. Pure Appl. Math. 60(12), 1769–1791 (2007)

  13. Caffarelli, L.A., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math. 155(3–4), 261–301 (1985)

  14. Crasta, G., Fragalà, I.: On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: regularity and geometric results. Arch. Ration. Mech. Anal. 218(3), 1577–1607 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Gabriel, R.M.: A result concerning convex level surfaces of 3-dimensional harmonic functions. J. Lond. Math. Soc. 32, 286–294 (1957)

    MathSciNet  MATH  Google Scholar 

  16. Guan, P., Ma, X.-N.: The Christoffel–Minkowski problem. I. Convexity of solutions of a Hessian equation. Invent. Math. 151(3), 553–577 (2003)

    MathSciNet  MATH  Google Scholar 

  17. Guan, P., Xu, L.: Convexity estimates for level sets of quasiconcave solutions to fully nonlinear elliptic equations. J. Reine Angew. Math. 680, 41–67 (2013)

    MathSciNet  MATH  Google Scholar 

  18. Guan, P., Ma, X.-N., Zhou, F.: The Christofel–Minkowski problem. III. Existence and convexity of admissible solutions. Commun. Pure Appl. Math. 59(9), 1352–1376 (2006)

    MathSciNet  MATH  Google Scholar 

  19. Hamel, F., Nadirashvili, N., Sire, Y.: Convexity of level sets for elliptic problems in convex domains or convex rings: two counterexamples. Am. J. Math. 138(2), 499–527 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Harvey, R., Lawson, H.B., Jr.: Calibrated geometries. Acta Math. 148, 47–157 (1982)

    MathSciNet  MATH  Google Scholar 

  21. Ishige, K., Nakagawa, K., Salani, P.: Power concavity in weakly coupled elliptic and parabolic systems. Nonlinear Anal. 131, 81–97 (2016)

    MathSciNet  MATH  Google Scholar 

  22. Jia, X., Ma, X.-N., Shi, S.: Remarks on convexity estimates for solutions of the torsion problem. To appear on Sci. China Math. (2023)

  23. Jia, X., Ma, X.-N., Shi, S.: Convexity estimates for Green’s function and the first eigenfunction of Laplace operator. To appear on Potential Anal. (2023)

  24. Kennington, A.U.: Power concavity and boundary value problems. Indiana Univ. Math. J. 34(3), 687–704 (1985)

    MathSciNet  MATH  Google Scholar 

  25. Korevaar, N.J.: Capillary surface convexity above convex domains. Indiana Univ. Math. J. 32(1), 73–81 (1983)

    MathSciNet  MATH  Google Scholar 

  26. Korevaar, N.J.: Convex solutions to nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 32(4), 603–614 (1983)

    MathSciNet  MATH  Google Scholar 

  27. Korevaar, N.J.: Convexity of level sets for solutions to elliptic ring problems. Commun. Partial Diff. Equ. 15(4), 541–556 (1990)

    MathSciNet  MATH  Google Scholar 

  28. Korevaar, N.J., Lewis, J.L.: Convex solutions of certain elliptic equations have constant rank Hessians. Arch. Rational Mech. Anal. 97(1), 19–32 (1987)

    MathSciNet  MATH  Google Scholar 

  29. Kulczycki, T.: On concavity of solutions of the Dirichlet problem for the equation \((-\Delta )^{1/2}\varphi =1\) in convex planar regions. J. Eur. Math. Soc. 19(5), 1361–1420 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Langford, M., Scheuer, J.: Concavity of solutions to degenerate elliptic equations on the sphere. Commun. Partial Diff. Equ. 46(6), 1005–1016 (2021)

    MathSciNet  MATH  Google Scholar 

  31. Lewis, J.L.: Capacitary functions in convex rings. Arch. Rational Mech. Anal. 66(3), 201–224 (1977)

    MathSciNet  MATH  Google Scholar 

  32. Liu, P., Ma, X.-N., Xu, L.: A Brunn–Minkowski inequality for the Hessian eigenvalue in three-dimensional convex domain. Adv. Math. 225(3), 1616–1633 (2010)

    MathSciNet  MATH  Google Scholar 

  33. Lu, S.: On the Dirichlet problem for Lagrangian phase equation with critical and supercritical phase. arXiv:2204.05420 (2022)

  34. Ma, X.-N., Xu, L.: The convexity of solution of a class Hessian equation in bounded convex domain in \(\mathbb{R} ^3\). J. Funct. Anal. 255(7), 1713–1723 (2008)

    MathSciNet  MATH  Google Scholar 

  35. Ma, X.-N., Zhang, W.: Superharmonicity of curvature function for the convex level sets of harmonic functions. Calc. Var. Part. Diff. Equ. 60(4), 141 (2021)

    MathSciNet  MATH  Google Scholar 

  36. Ma, X.-N., Ou, Q., Zhang, W.: Gaussian curvature estimates for the convex level sets of \(p\)-harmonic functions. Commun. Pure Appl. Math. 63(7), 935–971 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Ma, X.-N., Shi, S., Ye, Y.: The convexity estimates for the solutions of two elliptic equations. Commun. Partial Diff. Equ. 37(12), 2116–2137 (2012)

    MathSciNet  MATH  Google Scholar 

  38. Makar-Limanov, L.G.: The solution of the Dirichlet problem for the equation \(\Delta u=-1\) in a convex region. Mat. Zametki 9, 89–92 (1971)

    MathSciNet  MATH  Google Scholar 

  39. Nadirashvili, N., Vlăduţ, S.: Singular solution to special Lagrangian equations. Ann. Inst. H. Poincaré C 27(5), 1179–1188 (2010)

    MathSciNet  MATH  Google Scholar 

  40. Sakaguchi, S.: Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 14(3), 403–421 (1988)

    MathSciNet  MATH  Google Scholar 

  41. Salani, P.: Convexity of solutions and Brunn–Minkowski inequalities for Hessian equations in \(\mathbb{R}^{3}\). Adv. Math. 229(3), 1924–1948 (2012)

    MathSciNet  MATH  Google Scholar 

  42. Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory Second Expanded Edition. Encyclopedia of Mathematics and its Applications, vol. 151. Cambridge University Press, Cambridge (2014)

    Google Scholar 

  43. Székelyhidi, G., Weinkove, B.: On a constant rank theorem for nonlinear elliptic PDEs. Discrete Contin. Dyn. Syst. 36(11), 6523–6532 (2016)

    MathSciNet  MATH  Google Scholar 

  44. Székelyhidi, G., Weinkove, B.: Weak Harnack inequalities for eigenvalues and constant rank theorems. Commun. Partial Diff. Equ. 46(8), 1585–1600 (2021)

    MathSciNet  MATH  Google Scholar 

  45. Wang, X.-J.: Counterexample to the convexity of level sets of solutions to the mean curvature equation. J. Eur. Math. Soc. (JEMS) 16(6), 1173–1182 (2014)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  47. Wang, D., Yuan, Y.: Hessian estimates for special Lagrangian equations with critical and supercritical phases in general dimensions. Am. J. Math. 136(2), 481–499 (2014)

    MathSciNet  MATH  Google Scholar 

  48. Warren, M., Yuan, Y.: Explicit gradient estimates for minimal Lagrangian surfaces of dimension two. Math. Z. 262(4), 867–879 (2009)

    MathSciNet  MATH  Google Scholar 

  49. Warren, M., Yuan, Y.: Hessian and gradient estimates for three dimensional special Lagrangian equations with large phase. Am. J. Math. 132(3), 751–770 (2010)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first named author would like to thank Professor Xi-Nan Ma for several valuable discussions on this subject. Both the authors would like to thank the anonymous referees for carefully reading the manuscript and giving many useful comments which helped us improve the exposition of the paper. Research of the first named author was supported by the National Natural Science Foundation of China under Grants 11871255.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu Province, China

    Wei Zhang & Qi Zhou

Authors
  1. Wei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qi Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei Zhang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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

Zhang, W., Zhou, Q. Power Convexity of Solutions to a Special Lagrangian Equation in Dimension Two. J Geom Anal 33, 135 (2023). https://doi.org/10.1007/s12220-022-01186-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-022-01186-6

Keywords

Mathematics Subject Classification

Search

Navigation