### 2016年度　英語理学先端講義（数学3）   Lecture on Advanced Science in English (Mathematics 3)

メディア利用

クラス
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ZUA.E340

1

2016年度

3Q
シラバス更新日
2016年9月8日

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アクセスランキング media

### 講義の概要とねらい

This course is an introduction to the Monge-Ampere equation. After introducing basic concepts such as normal mapping and Monge-Ampere measure, we will study important properties of the Monge-Ampere measure including: weak continuity and invariance property. We then study the celebrated Aleksandrov's maximum principle, the comparison principle, John's lemma and applications. Finally, we will discuss the Dirichlet problem and sections of convex solutions to the Monge-Ampere equation.

The Monge-Ampere equation appears in many areas and applications including affine geometry, convex geometry, optimal transportation and meteorology. The Monge-Ampere equation and its applications is a very active area of research. This course hopes to provide solid background and motivate interested students entering this research area.

### 到達目標

・Be familiar with basic concepts in the Monge-Ampere equation such as normal mapping, Monge-Ampere measure, Aleksandrov's solution
・Be familiar with modern tools and concepts in the Monge-Ampere theory such as John's lemma and sections of solutions
・Understand and be able to use maximum principles in the Monge-Ampere equation

### キーワード

normal mapping, Monge-Ampere measure, weak continuity, invariance property, Aleksandrov's solution, Aleksandrov's maximum principle, comparison principle, John's lemma, Dirichlet problem, sections of convex functions.

### 学生が身につける力(ディグリー・ポリシー)

 ✔ 専門力 教養力 コミュニケーション力 展開力(探究力又は設定力) 展開力(実践力又は解決力)

### 授業の進め方

This is a standard lecture course. Homework will be assigned every week.

授業計画 課題

None required

### 参考書、講義資料等

The course will be based on Part 3 of the instructor's lecture notes “N. Q. Le, The second boundary value problem of the prescribed affine mean curvature equation and related linearized Monge-Ampere equation”, available at: http://pages.iu.edu/~nqle/SBVP_lectures.pdf

### 成績評価の基準及び方法

Final exam 50%, assignments 50%.

### 関連する科目

• MTH.C305 ： 実解析第一
• MTH.C351 ： 函数解析

### 履修の条件(知識・技能・履修済科目等)

Basic knowledge on advanced analysis is essential 