• Previous Article
    Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term
  • DCDS Home
  • This Issue
  • Next Article
    Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity
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 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, arXiv:1403.2133.

[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, arXiv:1409.3633.

[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, arXiv:1403.2133.

[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, arXiv:1409.3633.

[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.

[1]

Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013

[2]

Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247

[3]

Martino Bardi, Paola Mannucci. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2006, 5 (4) : 709-731. doi: 10.3934/cpaa.2006.5.709

[4]

Paola Mannucci. The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction. Communications on Pure and Applied Analysis, 2014, 13 (1) : 119-133. doi: 10.3934/cpaa.2014.13.119

[5]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601

[6]

Isabeau Birindelli, Francoise Demengel. The dirichlet problem for singluar fully nonlinear operators. Conference Publications, 2007, 2007 (Special) : 110-121. doi: 10.3934/proc.2007.2007.110

[7]

Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure and Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731

[8]

D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure and Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499

[9]

Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539

[10]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116

[11]

Michael Kühn. Power- and Log-concavity of viscosity solutions to some elliptic Dirichlet problems. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2773-2788. doi: 10.3934/cpaa.2018131

[12]

Marco Ghimenti, A. M. Micheletti. Non degeneracy for solutions of singularly perturbed nonlinear elliptic problems on symmetric Riemannian manifolds. Communications on Pure and Applied Analysis, 2013, 12 (2) : 679-693. doi: 10.3934/cpaa.2013.12.679

[13]

Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707

[14]

Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure and Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213

[15]

Hongxia Zhang, Ying Wang. Liouville results for fully nonlinear integral elliptic equations in exterior domains. Communications on Pure and Applied Analysis, 2018, 17 (1) : 85-112. doi: 10.3934/cpaa.2018006

[16]

Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019

[17]

Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771

[18]

Agnid Banerjee. A note on the unique continuation property for fully nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2015, 14 (2) : 623-626. doi: 10.3934/cpaa.2015.14.623

[19]

Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128

[20]

Jianguo Huang, Jun Zou. Uniform a priori estimates for elliptic and static Maxwell interface problems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 145-170. doi: 10.3934/dcdsb.2007.7.145

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (185)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]