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February  2016, 36(2): 701-714. doi: 10.3934/dcds.2016.36.701

The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds

Bo Guan 1, and  Heming Jiao 2,

1. 

Department of Mathematics, The Ohio State University, Columbus, OH 43210, United States
2. 

Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

Received  July 2014 Revised  January 2015 Published  August 2015

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.
Keywords: concavity., a priori estimates, Fully nonlinear elliptic equations on Riemannian manifolds, subsolutions, Dirichlet problem.
Mathematics Subject Classification: 35J15, 58J05, 35B4.
Citation: Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 701-714. doi: 10.3934/dcds.2016.36.701
References:
[1]

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

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

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

[5]

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.  Google Scholar

[6]

B. Guan, The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds,, preprint, ().   Google Scholar

[7]

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

[8]

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., ().   Google Scholar

[9]

B. Guan, S.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds,, preprint, ().   Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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.  Google Scholar

[14]

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

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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.  Google Scholar

show all references

References:
[1]

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

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

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

[5]

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.  Google Scholar

[6]

B. Guan, The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds,, preprint, ().   Google Scholar

[7]

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

[8]

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., ().   Google Scholar

[9]

B. Guan, S.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds,, preprint, ().   Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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.  Google Scholar

[14]

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

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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.  Google Scholar

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