Advanced Search
Article Contents
Article Contents

The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds

Abstract Related Papers Cited by
  • We apply some new ideas to derive $C^2$ estimates for solutions of a general class of fully nonlinear elliptic equations on Riemannian manifolds under a ``minimal'' set of assumptions which are standard in the literature. Based on these estimates we solve the Dirichlet problem using the continuity method and degree theory.
    Mathematics Subject Classification: 35J15, 58J05, 35B45.


    \begin{equation} \\ \end{equation}
  • [1]

    A. D. Alexandrov, Uniqueness theorems for surfaces in the large, I, Vestnik Leningrad. Univ., 11 (1956), 5-17.


    L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations III: Functions of eigenvalues of the Hessians, Acta Math., 155 (1985), 261-301.doi: 10.1007/BF02392544.


    S. Y. Cheng and S. T. Yau, On the regularity of the solution of the n-dimensional Minkowski problem, Comm. Pure Applied Math., 29 (1976), 495-516.doi: 10.1002/cpa.3160290504.


    S. S. Chern, Integral formulas for hypersurfaces in Euclidean space and their applications to uniqueness theorems, J. Math. Mech., 8 (1959), 947-955.


    B. Guan, Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds, Duke Math. J., 163 (2014), 1491-1524.doi: 10.1215/00127094-2713591.


    B. Guan, The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds, preprint, arXiv:1403.2133.


    B. Guan and P.-F. Guan, Closed hypersurfaces of prescribed curvatures, Ann. Math. (2), 156 (2002), 655-673.doi: 10.2307/3597202.


    B. Guan and H.-M. Jiao, Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds, to appear in Calc. Var. PDE.


    B. Guan, S.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds, preprint, arXiv:1409.3633.


    B. Guan and J. Spruck, Interior gradient estimates for solutions of prescribed curvature equations of parabolic type, Indiana Univ. Math. J., 40 (1991), 1471-1481.doi: 10.1512/iumj.1991.40.40066.


    P.-F. Guan, J.-F. Li and Y.-Y. Li, Hypersurfaces of prescribed curvature measures, Duke Math. J., 161 (2012), 1927-1942.doi: 10.1215/00127094-1645550.


    P.-F. Guan and Y.-Y. Li, $C^{1,1}$ Regularity for solutions of a problem of Alexandrov, Comm. Pure Applied Math., 50 (1997), 789-811.doi: 10.1002/(SICI)1097-0312(199708)50:8<789::AID-CPA4>3.0.CO;2-2.


    P.-F. Guan and X.-N. Ma, The Christoffel-Minkowski problem. I. Convexity of solutions of a Hessian equation, Invent. Math., 151 (2003), 553-577.doi: 10.1007/s00222-002-0259-2.


    N. J. Korevaar, A priori gradient bounds for solutions to elliptic Weingarten equations, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 4 (1987), 405-421.


    Y.-Y. Li, Interior gradient estimates for solutions of certain fully nonlinear elliptic equations, J. Diff. Equations, 90 (1991), 172-185.doi: 10.1016/0022-0396(91)90166-7.


    L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Applied Math., 6 (1953), 337-394.doi: 10.1002/cpa.3160060303.


    A. V. Pogorelov, Regularity of a convex surface with given Gaussian curvature, Mat. Sb., 31 (1952), 88-103.


    A. V. Pogorelov, The Minkowski Multidimentional Problem, Winston, Washington, 1978.


    N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. National Mech. Anal., 111 (1990), 153-179.doi: 10.1007/BF00375406.


    J. Urbas, Hessian equations on compact Riemannian manifolds, Nonlinear Problems in Mathematical Physics and Related Topics II, Kluwer/Plenum, New York, 2 (2002), 367-377.doi: 10.1007/978-1-4615-0701-7_20.

  • 加载中

Article Metrics

HTML views() PDF downloads(211) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint