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

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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

Weisong Dong ^1, , Tingting Wang ^1, and Gejun Bao ^1,

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

Received September 2015 Revised March 2016 Published July 2016

Abstract
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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.

Keywords: Hessian, Obstacle problem, fully nonlinear elliptic equations, A priori estimates, Riemannian manifolds..

Mathematics Subject Classification: Primary: 35B45, 35J60; Secondary: 58J05, 35D4.

Citation: 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

References:

[1]	B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations, 2 (1994), 151-171. doi: 10.1007/BF01191340.
[2]	G.-J. Bao, W.-S. Dong and H.-M. Jiao, Regularity for an obstacle problem of Hessian equations on Riemannian manifolds, J. Differential Equations, 258 (2015), 696-716. doi: 10.1016/j.jde.2014.10.001.
[3]	L. A. Caffarelli and R. McCann, Free boundaries in optimal transport and Monge-Ampère obstacle problems, Ann. of Math., 171 (2010), 673-730. doi: 10.4007/annals.2010.171.673.
[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, Acta Math., 155 (1985), 261-301. doi: 10.1007/BF02392544.
[5]	M. Crandall, H. Ishii and P. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.
[6]	L. C. Evans, Classical solutions of fully nonlinear, convex, second order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-363. doi: 10.1002/cpa.3160350303.
[7]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations Of Second Order, 2nd edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0.
[8]	C. Gerhardt, Hypersurfaces of prescribed mean curvature over obstacles, Math. Z., 133 (1973), 169-185.
[9]	C. Gerhardt, Closed Weingarten hypersurfaces in Riemannian manifolds, J. Differential Geom., 43 (1996), 612-641.
[10]	B. Guan, The Dirichlet problem for Hessian equations on Riemannian manifolds, Calc. Var. Partial Differential Equations, 8 (1999), 45-69. doi: 10.1007/s005260050116.
[11]	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.
[12]	B. Guan, The Dirichlet problem for fully nonlinear ellipitc equations on Riemannian manifolds,, preprint, ().
[13]	B. Guan, S.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds, Anal. PDE, 8 (2015), 1145-1164.. doi: 10.2140/apde.2015.8.1145.
[14]	B. Guan and H.-M. Jiao, The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds, Discrete Conti. Dyn. Syst., 36 (2016), 701-714. doi: 10.3934/dcds.2016.36.701.
[15]	E. Giusti, Superfici minime cartesiane con ostaeoli diseontinui, Arch. Ration. Mech. Anal., 35 (1969), 47-82.
[16]	H.-M. Jiao, $C^{1,1}$ regularity for an obstacle problem of Hessian equations on Riemannian manifolds, Proc. Amer. Math. Soc., 144 (2016), 3441-3453.
[17]	H.-M. Jiao and Y. Wang, The obstacle problem for Hessian equations on Riemannian manifolds, Nonlinear Anal., 95 (2014), 543-552.
[18]	N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations in a domain, Izvestia Math. Ser., 47 (1983), 75-108.
[19]	D. S. Kinderlehrer, Variational inequalities with lower dimensional obstacles, Israel J. Math., 10 (1971), 339-348.
[20]	D. S. Kinderlehrer, How a minimal surface leaves an obstacle, Acta Math., 130 (1973), 221-242.
[21]	K. Lee, The obstacle problem for Monge-Ampère equation, Comm. Partial Differential Equations, 26 (2001), 33-42. doi: 10.1081/PDE-100002244.
[22]	Y.-Y. Li, Degree theory for second order nonlinear elliptic operators and its applications, Comm. Partial Differential Equations, 14 (1989), 1541-1578. doi: 10.1080/03605308908820666.
[23]	J.-K. Liu and B. Zhou, An obstacle problem for a class of Monge-Ampère type functionals, J. Differential Equations, 254 (2013), 1306-1325. doi: 10.1016/j.jde.2012.10.017.
[24]	A. Oberman, The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc., 135 (2007), 1689-1694. doi: 10.1090/S0002-9939-07-08887-9.
[25]	A. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc., 363 (2011), 5871-5886. doi: 10.1090/S0002-9947-2011-05240-2.
[26]	O. Savin, A free boundary problem with optimal transportation, Comm. Pure Appl. Math., 57 (2004), 126-140. doi: 10.1002/cpa.3041.
[27]	O. Savin, The obstacle problem for Monge Ampere equation, Calc. Var. Partial Differential Equations, 22 (2005), 303-320. doi: 10.1007/s00526-004-0275-8.
[28]	N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Ration. Mech. Anal., 111 (1990), 153-179. doi: 10.1007/BF00375406.
[29]	J. Urbas, Hessian equations on compact Riemannian manifolds, in Nonlinear Problems in Mathematical Physics and Related Topics, II, Kluwer/Plenum, (2002), 367-377. doi: 10.1007/978-1-4615-0701-7_20.
[30]	J.-G. Xiong and J.-G. Bao, The obstacle problem for Monge-Ampère type equations in non-convex domains, Commun. Pure Appl. Anal., 10 (2011), 59-68. doi: 10.3934/cpaa.2011.10.59.

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References:

[1]	B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations, 2 (1994), 151-171. doi: 10.1007/BF01191340.
[2]	G.-J. Bao, W.-S. Dong and H.-M. Jiao, Regularity for an obstacle problem of Hessian equations on Riemannian manifolds, J. Differential Equations, 258 (2015), 696-716. doi: 10.1016/j.jde.2014.10.001.
[3]	L. A. Caffarelli and R. McCann, Free boundaries in optimal transport and Monge-Ampère obstacle problems, Ann. of Math., 171 (2010), 673-730. doi: 10.4007/annals.2010.171.673.
[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, Acta Math., 155 (1985), 261-301. doi: 10.1007/BF02392544.
[5]	M. Crandall, H. Ishii and P. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.
[6]	L. C. Evans, Classical solutions of fully nonlinear, convex, second order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-363. doi: 10.1002/cpa.3160350303.
[7]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations Of Second Order, 2nd edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0.
[8]	C. Gerhardt, Hypersurfaces of prescribed mean curvature over obstacles, Math. Z., 133 (1973), 169-185.
[9]	C. Gerhardt, Closed Weingarten hypersurfaces in Riemannian manifolds, J. Differential Geom., 43 (1996), 612-641.
[10]	B. Guan, The Dirichlet problem for Hessian equations on Riemannian manifolds, Calc. Var. Partial Differential Equations, 8 (1999), 45-69. doi: 10.1007/s005260050116.
[11]	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.
[12]	B. Guan, The Dirichlet problem for fully nonlinear ellipitc equations on Riemannian manifolds,, preprint, ().
[13]	B. Guan, S.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds, Anal. PDE, 8 (2015), 1145-1164.. doi: 10.2140/apde.2015.8.1145.
[14]	B. Guan and H.-M. Jiao, The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds, Discrete Conti. Dyn. Syst., 36 (2016), 701-714. doi: 10.3934/dcds.2016.36.701.
[15]	E. Giusti, Superfici minime cartesiane con ostaeoli diseontinui, Arch. Ration. Mech. Anal., 35 (1969), 47-82.
[16]	H.-M. Jiao, $C^{1,1}$ regularity for an obstacle problem of Hessian equations on Riemannian manifolds, Proc. Amer. Math. Soc., 144 (2016), 3441-3453.
[17]	H.-M. Jiao and Y. Wang, The obstacle problem for Hessian equations on Riemannian manifolds, Nonlinear Anal., 95 (2014), 543-552.
[18]	N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations in a domain, Izvestia Math. Ser., 47 (1983), 75-108.
[19]	D. S. Kinderlehrer, Variational inequalities with lower dimensional obstacles, Israel J. Math., 10 (1971), 339-348.
[20]	D. S. Kinderlehrer, How a minimal surface leaves an obstacle, Acta Math., 130 (1973), 221-242.
[21]	K. Lee, The obstacle problem for Monge-Ampère equation, Comm. Partial Differential Equations, 26 (2001), 33-42. doi: 10.1081/PDE-100002244.
[22]	Y.-Y. Li, Degree theory for second order nonlinear elliptic operators and its applications, Comm. Partial Differential Equations, 14 (1989), 1541-1578. doi: 10.1080/03605308908820666.
[23]	J.-K. Liu and B. Zhou, An obstacle problem for a class of Monge-Ampère type functionals, J. Differential Equations, 254 (2013), 1306-1325. doi: 10.1016/j.jde.2012.10.017.
[24]	A. Oberman, The convex envelope is the solution of a nonlinear obstacle problem, Proc. Amer. Math. Soc., 135 (2007), 1689-1694. doi: 10.1090/S0002-9939-07-08887-9.
[25]	A. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc., 363 (2011), 5871-5886. doi: 10.1090/S0002-9947-2011-05240-2.
[26]	O. Savin, A free boundary problem with optimal transportation, Comm. Pure Appl. Math., 57 (2004), 126-140. doi: 10.1002/cpa.3041.
[27]	O. Savin, The obstacle problem for Monge Ampere equation, Calc. Var. Partial Differential Equations, 22 (2005), 303-320. doi: 10.1007/s00526-004-0275-8.
[28]	N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Ration. Mech. Anal., 111 (1990), 153-179. doi: 10.1007/BF00375406.
[29]	J. Urbas, Hessian equations on compact Riemannian manifolds, in Nonlinear Problems in Mathematical Physics and Related Topics, II, Kluwer/Plenum, (2002), 367-377. doi: 10.1007/978-1-4615-0701-7_20.
[30]	J.-G. Xiong and J.-G. Bao, The obstacle problem for Monge-Ampère type equations in non-convex domains, Commun. Pure Appl. Anal., 10 (2011), 59-68. doi: 10.3934/cpaa.2011.10.59.

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