H28年度 英語理学先端講義(数学3)   Lecture on Advanced Science in English (Mathematics 3)

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開講元
数学科
担当教員名
利根川 吉廣 
授業形態
講義
曜日・時限(講義室)
集中講義等   
クラス
-
科目コード
ZUA.E340
単位数
1
開講年度
H28年度
開講クォーター
3Q
シラバス更新日
H28年9月8日
講義資料更新日
-
使用言語
英語
アクセスランキング

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

国際的教養力 コミュニケーション力 専門力 課題設定力 実践力または解決力
- - - -

授業の進め方

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

授業計画・課題

  授業計画 課題
第1回 The Monge-Ampere equation in different contexts; normal mapping, Monge-Ampere measure, Legendre transform Details will be provided during each class
第2回 Aleksandrov's solution, Examples, Weak continuity of Monge-Ampere measure, Details will be provided during each class
第3回 Invariances of the Monge-Ampere equation, maximum principles Details will be provided during each class
第4回 Aleksandrov's maximum principle Details will be provided during each class
第5回 John's lemma, comparison principle Details will be provided during each class
第6回 The Dirichlet's problem: uniqueness and solvability by the Perron method Details will be provided during each class
第7回 Sections of convex functions Details will be provided during each class
第8回 Geometric properties of sections of solutions to the Monge-Ampere equation Details will be provided during each class

教科書

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

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