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.
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_1291485 |
| Date | January 2015 |
| Contributors | Cheng, Siu Hong (author.), Lee, Woon Yin Paul (thesis advisor.), Chinese University of Hong Kong Graduate School. Division of Mathematics. (degree granting institution.) |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography, text |
| Format | electronic resource, electronic resource, remote, 1 online resource (81 leaves), computer, online resource |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons "Attribution-NonCommercial-NoDerivatives 4.0 International" License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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