The Neumann Problem for Hessian Equations

  • Xinan MaEmail author
  • Guohuan Qiu


In this paper, we prove the existence of a classical solution to a Neumann boundary value problem for Hessian equations in uniformly convex domain. The method depends upon the establishment of a priori derivative estimates up to second order. So we give an affirmative answer to a conjecture of N. Trudinger in 1987.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



Both authors would like to thank the helpful discussion and encouragement from Prof. P. Guan and Prof. X.-J.Wang. The first author would also thank Prof. N. Trudinger and Prof. J. Urbas for their interest and encouragement. The research of the first author was supported by NSFC 11471188 and NSFC 11721101. The second author was supported by the grant from USTC. Both authors were supported by Wu Wen-Tsun Key Lab.


  1. 1.
    Caffarelli L., Nirenberg L., Spruck J.: Dirichlet problem for nonlinear second order elliptic equations III, Functions of the eigenvalues of the Hessian. Acta Math. 155, 261–301 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Canic S., Keyfitz B., Lieberman G.: A proof of existence of perturbed steady transonic shocks via a free boundary problem. Commun. Pure Appl. Math. 53, 484–511 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, S., Chang, A.: On a fully non-linear elliptic PDE in conformal geometry. Proceedings conference in memory of Jose Fernando Escobar, Mat. Ense. Univ. (N.S.) 15(suppl. 1), 17–36 (2007)Google Scholar
  4. 4.
    Chen S.: Boundary value problems for some fully nonlinear elliptic equations. Cal. Var. Partial Differ. Equ. 30, 1–15 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen S.: Conformal deformation on manifolds with boundary. Geom. Funct. Anal. 19, 1029–1064 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chou K.S., Wang X.-J.: A variational theory of the Hessian equation. Commun. Pure Appl. Math. 54, 1029–1064 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Finn R.: Equilibrium capillary surfaces. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  8. 8.
    Gerhardt C.: Global regularity of the solutions to the capillary problem. Ann. Scuola Norm. Super. Pisa Cl. Sci. (4)(3), 151–176 (1976)Google Scholar
  9. 9.
    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer, Berlin (1977). ISBN: 3-540-08007-4Google Scholar
  10. 10.
    Guan B.: Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds. Duke Math. J. 163(8), 1491–1524 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ivochkina, N.: Solutions of the Dirichlet problem for certain equations of Monge–Ampere type (in Russian). Mat. Sb., 128 (1985), 403–415: English translation in Math. USSR Sb. 56(1987)Google Scholar
  12. 12.
    Ivochkina N., Trudinger N., Wang X.-J.: The Dirichlet problem for degenerate Hessian equations. Commun. Partial Differ. Equ. 29, 219–235 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jin Q., Li A., Li Y.Y.: Estimates and existence results for a fully nonlinear Yamabe problem on manifolds with boundary. Cal. Var. Partial Differ. Equ. 28, 509–543 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lieberman G.: Second Order Parabolic Differential Equations. World scientific, Singapore (1996)CrossRefzbMATHGoogle Scholar
  15. 15.
    Lieberman G.: Oblique Boundary Value Problems for Elliptic Equations. World Scientific Publishing, Singapore (2013)CrossRefzbMATHGoogle Scholar
  16. 16.
    Lieberman G., Trudinger N.: Nonlinear oblique boundary value problems for nonlinear elliptic equations. Trans. Am. Math. Soc. 295(2), 509–546 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lin M., Trudinger N.: On some inequalities for elementary symmetric functions. Bull. Aust. Math. Soc. 50, 317–326 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lion P.L., Trudinger N.: Linear oblique derivative problems for the uniformly elliptic Hamilton–Jacobi–Bellman equation. Math. Zeit. 191, 1–15 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lions P.L., Trudinger N., Urbas J.: The Neumann problem for equations of Monge–Ampere type. Commun. Pure Appl. Math. 39, 539–563 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lott J.: Mean curvature flow in a Ricci flow background. Commun. Math. Phys. 313, 517–533 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ma X.N., Qiu G.H., Xu J.J.: Gradient estimates on Hessian equations for Neumann problem. Sci. Sin. Math. 46, 1117–1126 (2016) ChineseGoogle Scholar
  22. 22.
    Ma X.N., Xu J.J.: Gradient estimates of mean curvature equations with Neumann boundary condition. Adv. Math. 290, 1010–1039 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Simon L., Spruck J.: Existence and regularity of a capillary surface with prescribed contact angle. Arch. Ration. Mech. Anal. 61, 19–34 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Trudinger N.S.: On degenerate fully nonlinear elliptic equations in balls. Bull. Aust. Math. Soc. 35, 299–307 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Trudinger N.S.: On the Dirichlet problem for Hessian equations. Acta Math. 175, 151–164 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Trudinger N., Wang X.-J.: Hessian measures II. Ann. Math., (2) 150(2), 579–604 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ural’tseva, N.: Solvability of the capillary problem, Vestnik Leningrad. Univ. No. 19, 54–64 (1973), No. 1(1975), 143–149 [Russian]. English Translation in vestnik Leningrad Univ. Math. 6(1979), 363–375, 8(1980), 151–158Google Scholar
  28. 28.
    Urbas J.: Nonlinear oblique boundary value problems for Hessian equations in two dimensions. Ann. Inst. Henri Poincare Non Linear Anal. 12, 507–575 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Urbas J.: Nonlinear oblique boundary value problems for two-dimensional curvature equations. Adv. Differ. Equ. 1(3), 301–336 (1996)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Wang X.-J.: Oblique derivative problems for the equations of Monge–Ampere type. Chin. J. Contemp. Math. 13, 13–22 (1992)MathSciNetGoogle Scholar
  31. 31.
    Wang, X.-J.: The k-Hessian equation. In: Gursky, M. J., et al. (eds.) Geometric Analysis and PDEs (Cetraro, 2007), Lecture Notes in Mathematics. 1977, pp. 177–252. Springer, Dordrecht (2009)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China
  2. 2.Department of MathematicsThe Chinese University of Hong KongShatinPeople’s Republic of China

Personalised recommendations