Skip to main content
SpringerLink
Log in

Existence of Entire Solutions to the Lagrangian Mean Curvature Equations in Supercritical Phase

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

Abstract

In this paper, we establish the existence and uniqueness theorem of entire solutions to the Lagrangian mean curvature equations with prescribed asymptotic behavior at infinity. The phase functions are assumed to be supercritical and converge to a constant in a certain rate at infinity. The basic idea is to establish uniform estimates for the approximating problems defined on bounded domains and the main ingredient is to construct appropriate subsolutions and supersolutions as barrier functions. We also prove a nonexistence result to show the convergence rate of the phase functions is optimal.

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. Bao, J., Feng, Q.: Necessary and sufficient conditions on global solvability for the \(p\)-\(k\)-Hessian inequalities. Can. Math. Bull. 65(4), 1004–1019 (2022)

    Article  MathSciNet  Google Scholar 

  2. Bao, J., Li, H., Li, Y.: On the exterior Dirichlet problem for Hessian equations. Trans. Am. Math. Soc. 366(12), 6183–6200 (2014)

    Article  MathSciNet  Google Scholar 

  3. Bao, J., Li, H., Zhang, L.: Monge-Ampère equation on exterior domains. Calc. Var. Partial Differ. Equ. 52(1–2), 39–63 (2015)

    Article  Google Scholar 

  4. Bao, J., Li, H., Zhang, L.: Global solutions and exterior Dirichlet problem for Monge-Ampère equation in \({\mathbb{R} }^2\). Differ. Integral Equ. 29(5–6), 563–582 (2016)

    Google Scholar 

  5. Bao, J., Xiong, J., Zhou, Z.: Existence of entire solutions of Monge-Ampère equations with prescribed asymptotic behavior. Calc. Var. Partial Differ. Equ. 58(6), Paper No. 193, 12 (2019)

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

  7. Bhattacharya, A., Monney, C., Shankar, R.: Gradient estimates for the Lagrangian mean curvature equation with critical and supercritical phase. arXiv. 2205.13096 (2022)

  8. Bodine, S., Lutz, D.A.: Asymptotic Integration of Differential and Difference Equations. Lecture Notes in Mathematics, vol. 2129. Springer, Cham (2015)

    Google Scholar 

  9. Caffarelli, L., Li, Y.: An extension to a theorem of Jörgens, Calabi, and Pogorelov. Commun. Pure Appl. Math. 56(5), 549–583 (2003)

    Article  Google Scholar 

  10. Caffarelli, L., Li, Y.: A Liouville theorem for solutions of the Monge-Ampère equation with periodic data. Ann. l’Institut Henri Poincaré 21(1), 97–120 (2004)

    MathSciNet  Google Scholar 

  11. Caffarelli, L., Li, Y., Nirenberg, L.: Some remarks on singular solutions of nonlinear elliptic equations III: viscosity solutions including parabolic operators. Commun. Pure Appl. Math. 66(1), 109–143 (2013)

    Article  MathSciNet  Google Scholar 

  12. Caffarelli, L., 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)

    Article  MathSciNet  Google Scholar 

  13. Caffarelli, L.A., Tang, L., Wang, X.-J.: Global \(C^{1,\alpha }\) regularity for Monge-Ampère equation and convex envelope. Arch. Ration. Mech. Anal. 244(1), 127–155 (2022)

    Article  MathSciNet  Google Scholar 

  14. Capuzzo Dolcetta, I., Leoni, F., Vitolo, A.: Entire subsolutions of fully nonlinear degenerate elliptic equations. Bull. Inst. Math. Acad. Sin. New Ser. 9(2), 147–161 (2014)

    MathSciNet  Google Scholar 

  15. Capuzzo Dolcetta, I., Leoni, F., Vitolo, A.: On the inequality \(F(x, D^2u)\ge f(u)+g(u)\vert Du\vert ^q\). Math. Ann. 365(1–2), 423–448 (2016)

    Article  MathSciNet  Google Scholar 

  16. Earl, A.: Coddington and Norman Levinson. Theory of Ordinary Differential Equations, McGraw-Hill Book Company Inc, New York (1955)

    Google Scholar 

  17. Collins, T.C., Picard, S., Wu, X.: Concavity of the Lagrangian phase operator and applications. Calc. Var. Partial Differ. Equ. 56(4), Paper No. 89, 22 (2017)

  18. Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Am. Math. Soc. Bull. New Ser. 27(1), 1–67 (1992)

    Article  MathSciNet  Google Scholar 

  19. Esteban, M.J., Felmer, P.L., Quaas, A.: Superlinear elliptic equation for fully nonlinear operators without growth restrictions for the data. Proc. Edinb. Math. Soc. Ser. II 53(1), 125–141 (2010)

    Article  MathSciNet  Google Scholar 

  20. Felmer, P., Quaas, A., Sirakov, B.: Solvability of nonlinear elliptic equations with gradient terms. J. Differ. Equ. 254(11), 4327–4346 (2013)

    Article  MathSciNet  Google Scholar 

  21. Galise, G., Koike, S., Ley, O., Vitolo, A.: Entire solutions of fully nonlinear elliptic equations with a superlinear gradient term. J. Math. Anal. Appl. 441(1), 194–210 (2016)

    Article  MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

  24. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)

    Google Scholar 

  25. Ji, X., Bao, J.: Necessary and sufficient conditions on solvability for Hessian inequalities. Proc. Am. Math. Soc. 138(1), 175–188 (2010)

    Article  MathSciNet  Google Scholar 

  26. Jia, X.: Asymptotic behavior of solutions of fully nonlinear equations over exterior domains. C.R. Math. Acad. Sci. Paris 358(11–12), 1187–1197 (2020)

    MathSciNet  Google Scholar 

  27. Jia, X., Li, D.: The asymptotic behavior of viscosity solutions of Monge-Ampère equations in half space. Nonlinear Anal. 206, Paper No. 112229, 23 (2021)

  28. Jia, X., Li, D., Li, Z.: Asymptotic behavior at infinity of solutions of Monge-Ampère equations in half spaces. J. Differ. Equ. 269(1), 326–348 (2020)

    Article  Google Scholar 

  29. Keller, J.B.: On solutions of \(\Delta u=f(u)\). Commun. Pure Appl. Math. 10, 503–510 (1957)

    Article  Google Scholar 

  30. Li, D., Li, Z., Yuan, Y.: A Bernstein problem for special Lagrangian equations in exterior domains. Adv. Math. 361, 106927, 29 (2020)

  31. Li, X., Wang, C.: On the exterior Dirichlet problem for Hessian-type fully nonlinear elliptic equations. Commun. Contemp. Math. Paper No. 2250082 (2023)

  32. Li, Y., Lu, S.: Existence and nonexistence to exterior Dirichlet problem for Monge-Ampère equation. Calc. Var. Partial Differ. Equ. 57(6), Paper No. 161, 17 (2018)

  33. Li, Z.: On the exterior Dirichlet problem for special Lagrangian equations. Trans. Am. Math. Soc. 372(2), 889–924 (2019)

    Article  MathSciNet  Google Scholar 

  34. Liu, Z., Bao, J.: Asymptotic expansion at infinity of solutions of Monge-Ampère type equations. Nonlinear Anal. 212, Paper No. 112450, 17 (2021)

  35. Liu, Z., Bao, J.: Asymptotic expansion and optimal symmetry of minimal gradient graph equations in dimension 2. Commun. Contemp. Math. Paper No. 2150110, 25 (2022)

  36. Liu, Z., Bao, J.: Asymptotic expansion at infinity of solutions of special Lagrangian equations. J Geom. Anal. 32(3), Paper No. 90, 34 (2022)

  37. Liu, Z., Bao, J.: Asymptotic expansion of 2-dimensional gradient graph with vanishing mean curvature at infinity. Commun. Pure Appl. Anal. 21(9), 2911–2931 (2022)

    Article  MathSciNet  Google Scholar 

  38. Siyuan, L.: On the Dirichlet problem for Lagrangian phase equation with critical and supercritical phase. Discrete Contin. Dyn. Syst. 43(7), 2561–2575 (2023)

    Article  MathSciNet  Google Scholar 

  39. Osserman, R.: On the inequality \(\Delta u\ge f(u)\). Pac. J. Math. 7, 1641–1647 (1957)

    Article  Google Scholar 

  40. Savin, O.V.: Pointwise \(C^{2,\alpha }\) estimates at the boundary for the Monge-Ampère equation. J. Am. Math. Soc. 26(1), 63–99 (2013)

    Article  Google Scholar 

  41. Savin, O.V.: A localization theorem and boundary regularity for a class of degenerate Monge-Ampère equations. J. Differ. Equ. 256(2), 327–388 (2014)

    Article  Google Scholar 

  42. Trudinger, N.S., Wang, X.-J.: Boundary regularity for the Monge-Ampère and affine maximal surface equations. Ann. Math. Second Ser. 167(3), 993–1028 (2008)

    Article  Google Scholar 

  43. Wang, C., Huang, R., Bao, J.: On the second boundary value problem for Lagrangian mean curvature equation. Calc. Var. Partial Differ. Equ. 62(3), Paper No. 74 (2023)

  44. Yuan, Yu.: Global solutions to special Lagrangian equations. Proc. Am. Math. Soc. 134(5), 1355–1358 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Corresponding Author J. Bao is supported by the National Key Research and Development Program of China (No. 2020YFA0712900) and the Beijing Natural Science Foundation (No. 1222017). The second author Z. Liu is supported by the China Postdoctoral Science Foundation (No. 2022M720327). The third author C. Wang is supported by “the Fundamental Research Funds for the Central Universities” in UIBE (No. 23QD04).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Jiguang Bao

  2. Institute of Applied Mathematics, Department of Mathematics, School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing, 100124, China

    Zixiao Liu

  3. School of Statistics, University of International Business and Economics, Beijing, 100029, China

    Cong Wang

Authors
  1. Jiguang Bao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zixiao Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiguang Bao.

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

Bao, J., Liu, Z. & Wang, C. Existence of Entire Solutions to the Lagrangian Mean Curvature Equations in Supercritical Phase. J Geom Anal 34, 146 (2024). https://doi.org/10.1007/s12220-024-01589-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-024-01589-7

Keywords

Mathematics Subject Classification

Search

Navigation