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標題: Monge-Ampère型方程古典解的存在性
On The Existence of Classical Solutions of Monge-Ampère Type Equations
作者: 許勝富
Hsu, Sheng-Fu
關鍵字: Monge-Ampère equation
出版社: 應用數學系所
引用: References [1] I. Bakelman, Convex Analysis and Nonlinear Geometric Elliptic Equations, Springer- Verlag, 1994. [2] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983. [3] Cristian E. Guti`errez, The Monge-Amp`ere Equation, Birkh¨auser, 2001. [4] Jiakun Liu, Neil Trudinger, Xu-Jia Wang, Communications in partial differential equations, vol. 35, pp. 165-184, 2010 [5] X. N. Ma, N. S. Trudinger and X.-J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal. 177 [6] H.L. Royden,Real Analysis (3rd ed.), New York: Macmillan, 1988. [7] N. S. Trudinger, Lectures on nonlinear elliptic equations of second order, univ. of Tokyo, 1995 [8] Neil S. Trudinger, Recent developments in elliptic partial differential equations of Monge-Amp`ere type, Proceedings on the International Congress of Mathematicians: Madrid, August 22-30,2006 : invited lectures [9] Neil S. Trudinger and Xu-Jia Wang, The Monge-Amp`ere equations and its geometric applications, Handbook of Geometric Analysis, International Press, 2008, Vol. I, pp. 467-524 [10] Neil S. Trudinger, Xu-Jia Wang, On the second boundary value problem of Monge- Amp`ere type equations and optimal transportation, Annali della Scuola Normale Superiore di Pisa. Classe di scienze, Vol. 8, 2009 , 143-174 [11] J.Urbas , Mass transfer problems, lecture note, University of Bonn 1998 [12] Richard L. Wheeden. And Antoni Zygmund., Measure and Integral: an introduction to Real Analysis, New York: Marcel Dekker, 1977.
在這篇論文中,我們討論當Omega是在Rn 的uniformly convex domain 時,
下列的Dirichlet problem
det(D^2u-h|Du|^2I) = f(x) in Omega
u = 0 on boundary of Omega,
古典解的存在性以及唯一性。我們證明了當f, Omega
足夠smooth 以及當h 是足夠小的正數時,上述的邊界值問題有唯一的convex 古典解。

This works deals with the existence and uniqueness of classical solutions of the Dirichlet problem
det(D2u − h|Du|2I) = f(x) in Omega
u = 0 on boundary of Omega,
in a uniformly convex domain Omega of Rn. In this thesis, we prove the existence and uniqueness of classical solutions of the problem (P) for all
small h > 0 under some smooth conditions on f and Omega.
其他識別: U0005-2207201115552100
Appears in Collections:應用數學系所

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