# American Institute of Mathematical Sciences

September  2016, 15(5): 1769-1780. doi: 10.3934/cpaa.2016013

## A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds

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

Received  September 2015 Revised  March 2016 Published  July 2016

We are concerned with A priori estimates for the obstacle problem of a wide class of fully nonlinear equations on Riemannian manifolds. We use new techniques introduced by Bo Guan and derive new results for A priori second order estimates of its singular perturbation problem under fairly general conditions. By approximation, the existence of a $C^{1,1}$ viscosity solution is proved.
Citation: Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013
##### References:
 [1] B. Andrews, Contraction of convex hypersurfaces in Euclidean space,, \emph{Calc. Var. Partial Differential Equations}, 2 (1994), 151. doi: 10.1007/BF01191340. Google Scholar [2] G.-J. Bao, W.-S. Dong and H.-M. Jiao, Regularity for an obstacle problem of Hessian equations on Riemannian manifolds,, \emph{J. Differential Equations}, 258 (2015), 696. doi: 10.1016/j.jde.2014.10.001. Google Scholar [3] L. A. Caffarelli and R. McCann, Free boundaries in optimal transport and Monge-Ampère obstacle problems,, \emph{Ann. of Math.}, 171 (2010), 673. doi: 10.4007/annals.2010.171.673. Google Scholar [4] L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations III, Functions of the eigenvalues of the Hessian,, \emph{Acta Math.}, 155 (1985), 261. doi: 10.1007/BF02392544. Google Scholar [5] M. Crandall, H. Ishii and P. Lions, User's guide to viscosity solutions of second order partial differential equations,, \emph{Bull. Amer. Math. Soc.}, 27 (1992), 1. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar [6] L. C. Evans, Classical solutions of fully nonlinear, convex, second order elliptic equations,, \emph{Comm. Pure Appl. Math.}, 35 (1982), 333. doi: 10.1002/cpa.3160350303. Google Scholar [7] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations Of Second Order,, 2$^{nd}$ edition, (1983). doi: 10.1007/978-3-642-61798-0. Google Scholar [8] C. Gerhardt, Hypersurfaces of prescribed mean curvature over obstacles,, \emph{Math. Z.}, 133 (1973), 169. Google Scholar [9] C. Gerhardt, Closed Weingarten hypersurfaces in Riemannian manifolds,, \emph{J. Differential Geom.}, 43 (1996), 612. Google Scholar [10] B. Guan, The Dirichlet problem for Hessian equations on Riemannian manifolds,, \emph{Calc. Var. Partial Differential Equations}, 8 (1999), 45. doi: 10.1007/s005260050116. Google Scholar [11] B. Guan, Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds,, \emph{Duke Math. J.}, 163 (2014), 1491. doi: 10.1215/00127094-2713591. Google Scholar [12] B. Guan, The Dirichlet problem for fully nonlinear ellipitc equations on Riemannian manifolds,, preprint, (). Google Scholar [13] B. Guan, S.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds,, \emph{Anal. PDE}, 8 (2015), 1145. doi: 10.2140/apde.2015.8.1145. Google Scholar [14] B. Guan and H.-M. Jiao, The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds,, \emph{Discrete Conti. Dyn. Syst.}, 36 (2016), 701. doi: 10.3934/dcds.2016.36.701. Google Scholar [15] E. Giusti, Superfici minime cartesiane con ostaeoli diseontinui,, \emph{Arch. Ration. Mech. Anal.}, 35 (1969), 47. Google Scholar [16] H.-M. Jiao, $C^{1,1}$ regularity for an obstacle problem of Hessian equations on Riemannian manifolds,, \emph{Proc. Amer. Math. Soc.}, 144 (2016), 3441. Google Scholar [17] H.-M. Jiao and Y. Wang, The obstacle problem for Hessian equations on Riemannian manifolds,, \emph{Nonlinear Anal.}, 95 (2014), 543. Google Scholar [18] N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations in a domain,, \emph{Izvestia Math. Ser.}, 47 (1983), 75. Google Scholar [19] D. S. Kinderlehrer, Variational inequalities with lower dimensional obstacles,, \emph{Israel J. Math.}, 10 (1971), 339. Google Scholar [20] D. S. Kinderlehrer, How a minimal surface leaves an obstacle,, \emph{Acta Math.}, 130 (1973), 221. Google Scholar [21] K. Lee, The obstacle problem for Monge-Ampère equation,, \emph{Comm. Partial Differential Equations}, 26 (2001), 33. doi: 10.1081/PDE-100002244. Google Scholar [22] Y.-Y. Li, Degree theory for second order nonlinear elliptic operators and its applications,, \emph{Comm. Partial Differential Equations}, 14 (1989), 1541. doi: 10.1080/03605308908820666. Google Scholar [23] J.-K. Liu and B. Zhou, An obstacle problem for a class of Monge-Ampère type functionals,, \emph{J. Differential Equations}, 254 (2013), 1306. doi: 10.1016/j.jde.2012.10.017. Google Scholar [24] A. Oberman, The convex envelope is the solution of a nonlinear obstacle problem,, \emph{Proc. Amer. Math. Soc.}, 135 (2007), 1689. doi: 10.1090/S0002-9939-07-08887-9. Google Scholar [25] A. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope,, \emph{Trans. Amer. Math. Soc.}, 363 (2011), 5871. doi: 10.1090/S0002-9947-2011-05240-2. Google Scholar [26] O. Savin, A free boundary problem with optimal transportation,, \emph{Comm. Pure Appl. Math.}, 57 (2004), 126. doi: 10.1002/cpa.3041. Google Scholar [27] O. Savin, The obstacle problem for Monge Ampere equation,, \emph{Calc. Var. Partial Differential Equations}, 22 (2005), 303. doi: 10.1007/s00526-004-0275-8. Google Scholar [28] N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations,, \emph{Arch. Ration. Mech. Anal.}, 111 (1990), 153. doi: 10.1007/BF00375406. Google Scholar [29] J. Urbas, Hessian equations on compact Riemannian manifolds,, in \emph{Nonlinear Problems in Mathematical Physics and Related Topics, (2002), 367. doi: 10.1007/978-1-4615-0701-7_20. Google Scholar [30] J.-G. Xiong and J.-G. Bao, The obstacle problem for Monge-Ampère type equations in non-convex domains,, \emph{Commun. Pure Appl. Anal.}, 10 (2011), 59. doi: 10.3934/cpaa.2011.10.59. Google Scholar

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##### References:
 [1] B. Andrews, Contraction of convex hypersurfaces in Euclidean space,, \emph{Calc. Var. Partial Differential Equations}, 2 (1994), 151. doi: 10.1007/BF01191340. Google Scholar [2] G.-J. Bao, W.-S. Dong and H.-M. Jiao, Regularity for an obstacle problem of Hessian equations on Riemannian manifolds,, \emph{J. Differential Equations}, 258 (2015), 696. doi: 10.1016/j.jde.2014.10.001. Google Scholar [3] L. A. Caffarelli and R. McCann, Free boundaries in optimal transport and Monge-Ampère obstacle problems,, \emph{Ann. of Math.}, 171 (2010), 673. doi: 10.4007/annals.2010.171.673. Google Scholar [4] L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations III, Functions of the eigenvalues of the Hessian,, \emph{Acta Math.}, 155 (1985), 261. doi: 10.1007/BF02392544. Google Scholar [5] M. Crandall, H. Ishii and P. Lions, User's guide to viscosity solutions of second order partial differential equations,, \emph{Bull. Amer. Math. Soc.}, 27 (1992), 1. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar [6] L. C. Evans, Classical solutions of fully nonlinear, convex, second order elliptic equations,, \emph{Comm. Pure Appl. Math.}, 35 (1982), 333. doi: 10.1002/cpa.3160350303. Google Scholar [7] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations Of Second Order,, 2$^{nd}$ edition, (1983). doi: 10.1007/978-3-642-61798-0. Google Scholar [8] C. Gerhardt, Hypersurfaces of prescribed mean curvature over obstacles,, \emph{Math. Z.}, 133 (1973), 169. Google Scholar [9] C. Gerhardt, Closed Weingarten hypersurfaces in Riemannian manifolds,, \emph{J. Differential Geom.}, 43 (1996), 612. Google Scholar [10] B. Guan, The Dirichlet problem for Hessian equations on Riemannian manifolds,, \emph{Calc. Var. Partial Differential Equations}, 8 (1999), 45. doi: 10.1007/s005260050116. Google Scholar [11] B. Guan, Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds,, \emph{Duke Math. J.}, 163 (2014), 1491. doi: 10.1215/00127094-2713591. Google Scholar [12] B. Guan, The Dirichlet problem for fully nonlinear ellipitc equations on Riemannian manifolds,, preprint, (). Google Scholar [13] B. Guan, S.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds,, \emph{Anal. PDE}, 8 (2015), 1145. doi: 10.2140/apde.2015.8.1145. Google Scholar [14] B. Guan and H.-M. Jiao, The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds,, \emph{Discrete Conti. Dyn. Syst.}, 36 (2016), 701. doi: 10.3934/dcds.2016.36.701. Google Scholar [15] E. Giusti, Superfici minime cartesiane con ostaeoli diseontinui,, \emph{Arch. Ration. Mech. Anal.}, 35 (1969), 47. Google Scholar [16] H.-M. Jiao, $C^{1,1}$ regularity for an obstacle problem of Hessian equations on Riemannian manifolds,, \emph{Proc. Amer. Math. Soc.}, 144 (2016), 3441. Google Scholar [17] H.-M. Jiao and Y. Wang, The obstacle problem for Hessian equations on Riemannian manifolds,, \emph{Nonlinear Anal.}, 95 (2014), 543. Google Scholar [18] N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations in a domain,, \emph{Izvestia Math. Ser.}, 47 (1983), 75. Google Scholar [19] D. S. Kinderlehrer, Variational inequalities with lower dimensional obstacles,, \emph{Israel J. Math.}, 10 (1971), 339. Google Scholar [20] D. S. Kinderlehrer, How a minimal surface leaves an obstacle,, \emph{Acta Math.}, 130 (1973), 221. Google Scholar [21] K. Lee, The obstacle problem for Monge-Ampère equation,, \emph{Comm. Partial Differential Equations}, 26 (2001), 33. doi: 10.1081/PDE-100002244. Google Scholar [22] Y.-Y. Li, Degree theory for second order nonlinear elliptic operators and its applications,, \emph{Comm. Partial Differential Equations}, 14 (1989), 1541. doi: 10.1080/03605308908820666. Google Scholar [23] J.-K. Liu and B. Zhou, An obstacle problem for a class of Monge-Ampère type functionals,, \emph{J. Differential Equations}, 254 (2013), 1306. doi: 10.1016/j.jde.2012.10.017. Google Scholar [24] A. Oberman, The convex envelope is the solution of a nonlinear obstacle problem,, \emph{Proc. Amer. Math. Soc.}, 135 (2007), 1689. doi: 10.1090/S0002-9939-07-08887-9. Google Scholar [25] A. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope,, \emph{Trans. Amer. Math. Soc.}, 363 (2011), 5871. doi: 10.1090/S0002-9947-2011-05240-2. Google Scholar [26] O. Savin, A free boundary problem with optimal transportation,, \emph{Comm. Pure Appl. Math.}, 57 (2004), 126. doi: 10.1002/cpa.3041. Google Scholar [27] O. Savin, The obstacle problem for Monge Ampere equation,, \emph{Calc. Var. Partial Differential Equations}, 22 (2005), 303. doi: 10.1007/s00526-004-0275-8. Google Scholar [28] N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations,, \emph{Arch. Ration. Mech. Anal.}, 111 (1990), 153. doi: 10.1007/BF00375406. Google Scholar [29] J. Urbas, Hessian equations on compact Riemannian manifolds,, in \emph{Nonlinear Problems in Mathematical Physics and Related Topics, (2002), 367. doi: 10.1007/978-1-4615-0701-7_20. Google Scholar [30] J.-G. Xiong and J.-G. Bao, The obstacle problem for Monge-Ampère type equations in non-convex domains,, \emph{Commun. Pure Appl. Anal.}, 10 (2011), 59. doi: 10.3934/cpaa.2011.10.59. Google Scholar
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