The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds

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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

Abstract
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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 and 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.
[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.
[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.
[4]	S. S. Chern, Integral formulas for hypersurfaces in Euclidean space and their applications to uniqueness theorems, J. Math. Mech., 8 (1959), 947-955.
[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.
[6]	B. Guan, The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds,, preprint, ().
[7]	B. Guan and P.-F. Guan, Closed hypersurfaces of prescribed curvatures, Ann. Math. (2), 156 (2002), 655-673. doi: 10.2307/3597202.
[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., ().
[9]	B. Guan, S.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds,, preprint, ().
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	A. V. Pogorelov, Regularity of a convex surface with given Gaussian curvature, Mat. Sb., 31 (1952), 88-103.
[18]	A. V. Pogorelov, The Minkowski Multidimentional Problem, Winston, Washington, 1978.
[19]	N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. National Mech. Anal., 111 (1990), 153-179. doi: 10.1007/BF00375406.
[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.

show all references

References:

[1]	A. D. Alexandrov, Uniqueness theorems for surfaces in the large, I, Vestnik Leningrad. Univ., 11 (1956), 5-17.
[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.
[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.
[4]	S. S. Chern, Integral formulas for hypersurfaces in Euclidean space and their applications to uniqueness theorems, J. Math. Mech., 8 (1959), 947-955.
[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.
[6]	B. Guan, The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds,, preprint, ().
[7]	B. Guan and P.-F. Guan, Closed hypersurfaces of prescribed curvatures, Ann. Math. (2), 156 (2002), 655-673. doi: 10.2307/3597202.
[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., ().
[9]	B. Guan, S.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds,, preprint, ().
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	A. V. Pogorelov, Regularity of a convex surface with given Gaussian curvature, Mat. Sb., 31 (1952), 88-103.
[18]	A. V. Pogorelov, The Minkowski Multidimentional Problem, Winston, Washington, 1978.
[19]	N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. National Mech. Anal., 111 (1990), 153-179. doi: 10.1007/BF00375406.
[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.

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