Acta mathematica scientia, Series B >
WEAK SOLUTIONS OF MONGE-AMPÈRE TYPE EQUATIONS IN OPTIMAL TRANSPORTATION
Received date: 2012-06-15
Revised date: 2012-09-15
Online published: 2013-07-20
Supported by
This work was supported by National Natural Science Foundation of China (11071119).
This paper concerns the weak solutions of some Monge-Amp`ere type equa-tions in the optimal transportation theory. The relationship between the Aleksandrov solutions and the viscosity solutions of the Monge-Amp`ere type equations is discussed. A uniform estimate for solution of the Dirichlet problem with homogeneous boundary value is obtained.
Jiang Fei-Da , YANG Xiao-Ping . WEAK SOLUTIONS OF MONGE-AMPÈRE TYPE EQUATIONS IN OPTIMAL TRANSPORTATION[J]. Acta mathematica scientia, Series B, 2013 , 33(4) : 950 -962 . DOI: 10.1016/S0252-9602(13)60054-5
[1] Bakelman I J. Convex Analysis and Nonlinear Geometric Elliptic Equations. Berlin, Heidelberg, New York: Springer, 1994
[2] Brenier Y. Decomposition polaire et rearrangement monotone des champs de vecteurs. (French) C R Acad Sci Paris S`erie I Math, 1987, 305(19): 805–808
[3] Caffarelli L A. Interior W2,p estimates for solutions to the Monge-Amp`ere equation and their strict convexity. Ann Math, 1990, 131: 129–134
[4] Caffarelli L A, Cabr´e X. Fully Nonlinear Elliptic Equations. Colloquium Publications 43. Providence RI: Amer Math Sci, 1995
[5] Caffarelli L A, Nirenberg L, Spruck J. The Dirichlet problem for nonlinear second order elliptic eqations I: Monge-Amp`ere equations. Comm Pure Appl Math, 1984, 37: 369–402
[6] Crandall M G, Evans L C, Lions P L. Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans Amer Math Soc, 1984, 282: 487–502
[7] Crandall M G, Lions P L. Viscosity solutions of Hamilton-Jacobi equations. Trans Amer Math Soc, 1983, 277: 1–42
[8] Crandall M G, Ishii H, Lions P L. User’s guide to viscosity solutions of second order partial differential equations. Bulletin Amer Math Soc, 1992, 27(1): 1–67
[9] Gangbo W, McCann R J. The geometry of optimal transportation. Acta Math, 1996, 177(2): 113–116
[10] Guti´errez C E. The Monge-Amp`ere Equation. Boston, Basel, Berlin: Birkha¨user, 2001
[11] Guti´errez C E, Nguyen T V. On Monge-Amp`ere type equations arising in optimal transportation problems. Calc Var PDE, 2007, 28, 275–316
[12] Guti´errez C E, Tournier F. An Aleksandrov type estimate for -convex functions. Proc Amer Math Soc, 2010, 138(6): 2001–2014
[13] Jian H, Ju H, Liu Y, SunW. Symmetry of translating solutions to mean curvature flows. Acta Mathematica Scientia, 2010, 30B(6): 2006–2016
[14] Jiang F, Trudinger N S, Yang X P. On the Dirichlet problem for Monge-Amp`ere type equations. Preprint
[15] Liu J. Monge-Amp`ere Type Equations and Optimal Transportation [D]. Australian National University, 2010
[16] Loeper G. On the regularity of solutions of optimal transportation problems. Acta Math, 2009, 202: 241–283
[17] Ma X N, Trudinger N S,Wang X J. Regularity of potential functions of the optimal transportation problem. Arch Rat Mech Anal, 2005, 177: 151–183
[18] Trudinger N S, Wang X J. On the second boundary value problem for Monge-Amp`ere type equations and optimal transportation. Ann Scuola Norm Sup Pisa Cl Sci, 2009, 8: 143–174
[19] Villani C. Topics in Mass Transportation. Graduate Studies in Mathematics 58. Providence RI: Amer Math Soc, 2003
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