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

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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

Abstract
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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 and 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.
[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.
[3]	L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," Mathematical Society Colloquium Publications, 43. Amer. Math. Soc., Providence, RI, 1995.
[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.
[5]	L. Caffarelli and R. McCann, Free boundaries in optimal transport and Monge-Ampère obstacle problems, Ann. of Math., 171 (2010), 673-730.
[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.
[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.
[8]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Diiferential Equations of Second Order," Second Edition, Springer, Berlin, 1983.
[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.
[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.
[11]	B. Guan and J. Spruck, Boundary value problem on $\mathbb{S}^{n}$ for surfaces of constant Gauss curvature, Ann. of Math., 138 (1993), 601-624. doi: doi:10.2307/2946558.
[12]	C. Gutiérrez, "The Monge-Ampère equation,'', Progress in Nonlinear Differential Equations and their Applications, 44, Birkhäuser, Boston, 2001.
[13]	K. Lee, The obstacle problem for Monge-Ampère equation, Comm. Partial Diff. Eqn., 26 (2001), 33-42. doi: doi:10.1081/PDE-100002244.
[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.
[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.
[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.
[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.

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.
[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.
[3]	L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," Mathematical Society Colloquium Publications, 43. Amer. Math. Soc., Providence, RI, 1995.
[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.
[5]	L. Caffarelli and R. McCann, Free boundaries in optimal transport and Monge-Ampère obstacle problems, Ann. of Math., 171 (2010), 673-730.
[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.
[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.
[8]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Diiferential Equations of Second Order," Second Edition, Springer, Berlin, 1983.
[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.
[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.
[11]	B. Guan and J. Spruck, Boundary value problem on $\mathbb{S}^{n}$ for surfaces of constant Gauss curvature, Ann. of Math., 138 (1993), 601-624. doi: doi:10.2307/2946558.
[12]	C. Gutiérrez, "The Monge-Ampère equation,'', Progress in Nonlinear Differential Equations and their Applications, 44, Birkhäuser, Boston, 2001.
[13]	K. Lee, The obstacle problem for Monge-Ampère equation, Comm. Partial Diff. Eqn., 26 (2001), 33-42. doi: doi:10.1081/PDE-100002244.
[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.
[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.
[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.
[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.

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