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 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.

show all references

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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