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The existence of weak solutions for a generalized Camassa-Holm equation
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 |
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 $\mathbbS^n$ 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 $\mathbbS^n$ 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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