Th. Etat--Lettres--Paris--Faculté des lettres, 1951. / Tableau chronologique de l'oeuvre de Monge p. [377]-393. Bibliogr. p. [395]-425. Index.

Thesis (Ph. D.)--University of Washington, 2008. / Includes bibliographical references (p. 98-102).

The purpose of this thesis is to study the link between the theory of optimal transportation and the fully non-linear elliptic equation of the form [with formula] called the Monge-Ampère equation, where Ω is a bounded domain in Rⁿ. This equation is related to the optimal transportation problem associated with the quadratic cost c(x; y) = x.y, or equivalently the distance squared cost c(x,y) =1/2|x-y|². / The thesis consists of two parts. The first part is a summary of the classical theory about the optimal transportation problem proposed by Monge and Kantorovich, followed by the recent development pioneered by Ma, Trudinger, and Wang on the regularity of solutions. The Monge-Ampère equation satisfied by the solution of the Monge-Kantorovich problem will also be derived. / 本文的目的是研究最佳的運輸理論和定義在Rⁿ上的一個有界域，寫成[附圖]的Monge–Ampère 方程，兩者之間的關繫。這條屬於完全非線性橢圓方程，對於Monge 與Kantorovich 所提出的運輸數學問題中，代入二次函數作為成本函數時所推導出的偏微分方程。 / 本文由兩部分組成。第一部分是總結Monge 與Kantorovich 對於優化運輸數學所作出的貢獻，其後是論述偏微分方程學家Trudinger 與馬氏、王氏對於這個問題所作出的突破由他們的理論中，可以推導上出述的Monge–Ampère 方程。 / 第二部分是顯示第二邊值問題的適定性（存在解和方程解的唯一性）。為了有一個全面的學習，我們首先重溫橢圓方程的古典Schauder 理論。證明的核心部分是推導出邊界條件的傾斜度估計和對二階導數的估計，根據Urbas 所論證的方法。然後應用由Evans 和Krylov 兩者曾證明了有關完全非線性橢圓方程的定理，獲得對二階導數的Schauder 估算。我們證明偏微分方程存在解是運用連續性的方法。最後，我們將討論如何應用二階線性橢圓方程的定理獲得方程解的高階規律性。 / Cheng, Siu Hong. / Thesis M.Phil. Chinese University of Hong Kong 2015. / Includes bibliographical references (leaves 79-81). / Abstracts also in Chinese. / Title from PDF title page (viewed on 06, October, 2016). / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.

Thesis (Ph. D.)--Georgia Institute of Technology, 2005. / Includes bibliographical references (p. 108-110). Also available online via the Georgia Institute of Technology, website (http://etd.gatech.edu/).

In this dissertation we applied geometric methods to study underdetermined second order scalar ordinary differential equations (called general Monge equations), nonlinear involutive systems of two scalar partial differential equations in two independent variables and one unknown and non-Monge-Ampere Goursat parabolic scalar PDE in the plane. These particular kinds of differential equations are related to general rank-3 Pfaffian systems in five variables. Cartan studied these objects in his 1910 paper. In this work Cartan provided normal forms only for some general rank-3 Pfaffian systems with 14-, 7-, and 6-dimensional symmetry algebra. We applied our normal forms to [i] sharpen Cartan's integration method of nonlinear involutive systems, [ii] classify all general Monge equations with a freely acting transverse 3-dimensional symmetry algebra, of which many new examples are presented, and [iii] provide a broad classification of non-Monge-Ampere Darboux integrable hyperbolic PDE in the plane. We developed a computer software, called FiveVariables, that classifies general rank-3 Pfaffian systems. FiveVariables runs in the environment DifferentialGeometry of Maple, version 11 and later.

Apresentamos o problema do transporte ótimo de Monge-Kantorovich com duas medidas de probabilidade conhecidas e que possuem suporte em um conjunto de cardinalidade finita. O objetivo é determinar condições que permitam construir um acoplamento destas medidas que minimiza o valor esperado de uma função de custo conhecida e que assume valor nulo apenas nos elementos da diagonal. Apresentamos também um resultado relacionado com a solução do problema de Monge-Kantorovich em espaços produto finitos quando conhecemos soluções para o problema nos espaços marginais. / We present the Monge-Kantorovich optimal problem with two known probability measures on a finite set. The objective is to obtain conditions that allow us to build a coupling of these measures that minimizes the expected value of a cost function that is known and is zero only on the diagonal elements. We also present a result that is related with the solution of the Monge-Kantorovich problem in finite product spaces in the case that solutions to the problem in the marginal spaces are known.

Apresentamos o problema do transporte ótimo de Monge-Kantorovich com duas medidas de probabilidade conhecidas e que possuem suporte em um conjunto de cardinalidade finita. O objetivo é determinar condições que permitam construir um acoplamento destas medidas que minimiza o valor esperado de uma função de custo conhecida e que assume valor nulo apenas nos elementos da diagonal. Apresentamos também um resultado relacionado com a solução do problema de Monge-Kantorovich em espaços produto finitos quando conhecemos soluções para o problema nos espaços marginais. / We present the Monge-Kantorovich optimal problem with two known probability measures on a finite set. The objective is to obtain conditions that allow us to build a coupling of these measures that minimizes the expected value of a cost function that is known and is zero only on the diagonal elements. We also present a result that is related with the solution of the Monge-Kantorovich problem in finite product spaces in the case that solutions to the problem in the marginal spaces are known.

<p>In this thesis we study approximation of negative plurisubharmonic functions by functions defined on strictly larger domains. We show that, under certain conditions, every function <i>u</i> that is defined on a bounded hyperconvex domain Ω in C<i>n</i><i> </i>and has essentially boundary values zero and bounded Monge-Ampère mass, can be approximated by an increasing sequence of functions {<i>u</i><i>j</i>} that are defined on strictly larger domains, has boundary values zero and bounded Monge-Ampère mass. We also generalize this and show that, under the same conditions, the approximation property is true if the function u has essentially boundary values G, where G is a plurisubharmonic functions with certain properties. To show these approximation theorems we use subextension. We show that if Ω_1 and Ω_2 are hyperconvex domains in C<i>n</i> and if u is a plurisubharmonic function on Ω_1 with given boundary values and with bounded Monge-Ampère mass, then we can find a plurisubharmonic function û defined on Ω_2, with given boundary values, such that û <= u on Ω and with control over the Monge-Ampère mass of û.</p>

In this thesis we study approximation of negative plurisubharmonic functions by functions defined on strictly larger domains. We show that, under certain conditions, every function u that is defined on a bounded hyperconvex domain Ω in Cn and has essentially boundary values zero and bounded Monge-Ampère mass, can be approximated by an increasing sequence of functions {uj} that are defined on strictly larger domains, has boundary values zero and bounded Monge-Ampère mass. We also generalize this and show that, under the same conditions, the approximation property is true if the function u has essentially boundary values G, where G is a plurisubharmonic functions with certain properties. To show these approximation theorems we use subextension. We show that if Ω_1 and Ω_2 are hyperconvex domains in Cn and if u is a plurisubharmonic function on Ω_1 with given boundary values and with bounded Monge-Ampère mass, then we can find a plurisubharmonic function û defined on Ω_2, with given boundary values, such that û &lt;= u on Ω and with control over the Monge-Ampère mass of û.

In this thesis, we give a new definition of Hardy Spaces on hyperconvex domains in terms of Monge-Amp`ere measures which unifies the Hardy spaces on polydiscs and balls. Also we survey Monge-Amp`ere operators and Monge- Amp`ere measures.