American Institute of Mathematical Sciences

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

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.
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.

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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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