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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

 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.
Citation: Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 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.   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.  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.  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.   Google Scholar [5] B. Guan, Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds,, Duke Math. J., 163 (2014), 1491.  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.  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.  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.  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.  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.  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, 4 (1987), 405.   Google Scholar [15] Y.-Y. Li, Interior gradient estimates for solutions of certain fully nonlinear elliptic equations,, J. Diff. Equations, 90 (1991), 172.  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.  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.   Google Scholar [18] A. V. Pogorelov, The Minkowski Multidimentional Problem,, Winston, (1978).   Google Scholar [19] N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations,, Arch. National Mech. Anal., 111 (1990), 153.  doi: 10.1007/BF00375406.  Google Scholar [20] J. Urbas, Hessian equations on compact Riemannian manifolds,, Nonlinear Problems in Mathematical Physics and Related Topics II, 2 (2002), 367.  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.   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.  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.  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.   Google Scholar [5] B. Guan, Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds,, Duke Math. J., 163 (2014), 1491.  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.  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.  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.  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.  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.  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, 4 (1987), 405.   Google Scholar [15] Y.-Y. Li, Interior gradient estimates for solutions of certain fully nonlinear elliptic equations,, J. Diff. Equations, 90 (1991), 172.  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.  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.   Google Scholar [18] A. V. Pogorelov, The Minkowski Multidimentional Problem,, Winston, (1978).   Google Scholar [19] N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations,, Arch. National Mech. Anal., 111 (1990), 153.  doi: 10.1007/BF00375406.  Google Scholar [20] J. Urbas, Hessian equations on compact Riemannian manifolds,, Nonlinear Problems in Mathematical Physics and Related Topics II, 2 (2002), 367.  doi: 10.1007/978-1-4615-0701-7_20.  Google Scholar
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