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

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 & 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. 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.
[3]	L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,", Mathematical Society Colloquium Publications, (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. 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.
[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. 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. 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, (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. 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. doi: doi:10.1006/jdeq.1996.0174.
[11]	B. Guan and J. Spruck, Boundary value problem on $\mathbbS^n$ for surfaces of constant Gauss curvature,, Ann. of Math., 138 (1993), 601. doi: doi:10.2307/2946558.
[12]	C. Gutiérrez, "The Monge-Ampère equation,'', Progress in Nonlinear Differential Equations and their Applications, 44,, Birkh\, (2001).
[13]	K. Lee, The obstacle problem for Monge-Ampère equation,, Comm. Partial Diff. Eqn., 26 (2001), 33. 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. 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. 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. 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. 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. 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.
[3]	L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,", Mathematical Society Colloquium Publications, (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. 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.
[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. 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. 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, (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. 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. doi: doi:10.1006/jdeq.1996.0174.
[11]	B. Guan and J. Spruck, Boundary value problem on $\mathbbS^n$ for surfaces of constant Gauss curvature,, Ann. of Math., 138 (1993), 601. doi: doi:10.2307/2946558.
[12]	C. Gutiérrez, "The Monge-Ampère equation,'', Progress in Nonlinear Differential Equations and their Applications, 44,, Birkh\, (2001).
[13]	K. Lee, The obstacle problem for Monge-Ampère equation,, Comm. Partial Diff. Eqn., 26 (2001), 33. 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. 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. 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. 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. doi: doi:10.1007/BF00375406.

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