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January  2011, 10(1): 59-68. doi: 10.3934/cpaa.2011.10.59

The obstacle problem for Monge-Ampère type equations in non-convex domains

Jingang Xiong 1, and  Jiguang Bao 2,

1. 

School of Mathematical Sciences, Beijing Normal University, China
2. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875

Received  December 2009 Revised  March 2010 Published  November 2010

In this paper, we consider the obstacle problem for Monge-Ampère type equations which include prescribed Gauss curvature equation as a special case. We establish $C^{1,1}$ regularity of the greatest viscosity solution in non-convex domains.
Keywords: non-convex domain., Monge-Ampère equation, Obstacle problem.
Mathematics Subject Classification: Primary: 35J60; Secondary: 35R3.
Citation: Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59
References:
[1]

O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl., 76 (1997), 265-288. doi: doi:10.1016/S0021-7824(97)89952-7.  Google Scholar

[2]

J. Bao, The obstacle problems for second order fully nonlinear elliptic equations with Neumann boundary conditions, J. Partial Diff. Eqn., 3 (1992), 33-45.  Google Scholar

[3]

L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," Mathematical Society Colloquium Publications, 43. Amer. Math. Soc., Providence, RI, 1995.  Google Scholar

[4]

L. Caffarelli, A Localization property of viscosity solutions to the Monge-Ampere equation and their strict convexity, Ann. of Math., 131 (1990), 129-134. doi: doi:10.2307/1971509.  Google Scholar

[5]

L. Caffarelli and R. McCann, Free boundaries in optimal transport and Monge-Ampère obstacle problems, Ann. of Math., 171 (2010), 673-730.  Google Scholar

[6]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations I. Monge-Ampère equations, Comm. Pure Appl. Math., 37 (1984), 369-402. doi: doi:10.1002/cpa.3160370306.  Google Scholar

[7]

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: doi:10.1090/S0273-0979-1992-00266-5.  Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Diiferential Equations of Second Order," Second Edition, Springer, Berlin, 1983.  Google Scholar

[9]

B. Guan, The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature, Trans. Amer. Math. Soc., 350 (1998), 4955-4971. doi: doi:10.1090/S0002-9947-98-02079-0.  Google Scholar

[10]

B. Guan and Y. Y. Li, Monge-Ampère equations on Riemannian manifolds, J. Diff. Eqn., 132 (1996), 126-139. doi: doi:10.1006/jdeq.1996.0174.  Google Scholar

[11]

B. Guan and J. Spruck, Boundary value problem on $\mathbbS^n$ for surfaces of constant Gauss curvature, Ann. of Math., 138 (1993), 601-624. doi: doi:10.2307/2946558.  Google Scholar

[12]

C. Gutiérrez, "The Monge-Ampère equation,'', Progress in Nonlinear Differential Equations and their Applications, 44, Birkhäuser, Boston, 2001.  Google Scholar

[13]

K. Lee, The obstacle problem for Monge-Ampère equation, Comm. Partial Diff. Eqn., 26 (2001), 33-42. doi: doi:10.1081/PDE-100002244.  Google Scholar

[14]

Y. Y. Li, Some existence results of fully nonlinear elliptic equations of Monge-Ampère type, Comm. Pure Appl. Math., 43 (1990), 233-371. doi: doi:10.1002/cpa.3160430204.  Google Scholar

[15]

X. N. Ma, N. S. Trudinger and X-J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Rational Mech. Anal., 177 (2005), 151-183. doi: doi:10.1007/s00205-005-0362-9.  Google Scholar

[16]

O. Savin, The obstacle problem for Monge-Ampère equation, Calc. Var. Partial Diff. Eqn., 22 (2005), 303-320. doi: doi:10.1007/s00526-004-0275-8.  Google Scholar

[17]

N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal., 111 (1990), 153-179. doi: doi:10.1007/BF00375406.  Google Scholar

show all references

References:
[1]

O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl., 76 (1997), 265-288. doi: doi:10.1016/S0021-7824(97)89952-7.  Google Scholar

[2]

J. Bao, The obstacle problems for second order fully nonlinear elliptic equations with Neumann boundary conditions, J. Partial Diff. Eqn., 3 (1992), 33-45.  Google Scholar

[3]

L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," Mathematical Society Colloquium Publications, 43. Amer. Math. Soc., Providence, RI, 1995.  Google Scholar

[4]

L. Caffarelli, A Localization property of viscosity solutions to the Monge-Ampere equation and their strict convexity, Ann. of Math., 131 (1990), 129-134. doi: doi:10.2307/1971509.  Google Scholar

[5]

L. Caffarelli and R. McCann, Free boundaries in optimal transport and Monge-Ampère obstacle problems, Ann. of Math., 171 (2010), 673-730.  Google Scholar

[6]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations I. Monge-Ampère equations, Comm. Pure Appl. Math., 37 (1984), 369-402. doi: doi:10.1002/cpa.3160370306.  Google Scholar

[7]

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: doi:10.1090/S0273-0979-1992-00266-5.  Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Diiferential Equations of Second Order," Second Edition, Springer, Berlin, 1983.  Google Scholar

[9]

B. Guan, The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature, Trans. Amer. Math. Soc., 350 (1998), 4955-4971. doi: doi:10.1090/S0002-9947-98-02079-0.  Google Scholar

[10]

B. Guan and Y. Y. Li, Monge-Ampère equations on Riemannian manifolds, J. Diff. Eqn., 132 (1996), 126-139. doi: doi:10.1006/jdeq.1996.0174.  Google Scholar

[11]

B. Guan and J. Spruck, Boundary value problem on $\mathbbS^n$ for surfaces of constant Gauss curvature, Ann. of Math., 138 (1993), 601-624. doi: doi:10.2307/2946558.  Google Scholar

[12]

C. Gutiérrez, "The Monge-Ampère equation,'', Progress in Nonlinear Differential Equations and their Applications, 44, Birkhäuser, Boston, 2001.  Google Scholar

[13]

K. Lee, The obstacle problem for Monge-Ampère equation, Comm. Partial Diff. Eqn., 26 (2001), 33-42. doi: doi:10.1081/PDE-100002244.  Google Scholar

[14]

Y. Y. Li, Some existence results of fully nonlinear elliptic equations of Monge-Ampère type, Comm. Pure Appl. Math., 43 (1990), 233-371. doi: doi:10.1002/cpa.3160430204.  Google Scholar

[15]

X. N. Ma, N. S. Trudinger and X-J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Rational Mech. Anal., 177 (2005), 151-183. doi: doi:10.1007/s00205-005-0362-9.  Google Scholar

[16]

O. Savin, The obstacle problem for Monge-Ampère equation, Calc. Var. Partial Diff. Eqn., 22 (2005), 303-320. doi: doi:10.1007/s00526-004-0275-8.  Google Scholar

[17]

N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal., 111 (1990), 153-179. doi: doi:10.1007/BF00375406.  Google Scholar

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