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

show all references

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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