2021 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on current topics in Mathematics S
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Kazuhiro Ishige / Michiaki Onodera
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive
- Class
- -
- Course Code
- MTH.E648
- Number of credits
- 200
- Course offered
- 2021
- Offered quarter
- 4Q
- Syllabus updated
- Jul 10, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The main subject of this course is to understand power concavity properties of solutions to elliptic and parabolic equations. After introducing the notion of power concavity, we study the concavity maximum principle to obtain power concavity properties of solutions to elliptic and parabolic equations. Furthermore, we study non-preservation of quasi-concavity by the Dirichlet heat flow, preservation of log-concavity for parabolic equations, parabolic power concavity and their related topics.
Power concavity is a useful notion to describe the shape of solutions to partial differential equations, in particular, elliptic and parabolic equations. Thanks to the concavity maximum principle, power concavity properties for elliptic and parabolic equations have been studied by many mathematicians about for 40 years. Recently, power concavity properties were developed extensively via viscosity solutions. The aim of this lecture is to understand the outline of the recent development of power concavity properties of solutions to elliptic and parabolic equations.
Course description and aims
・Understand the definition of power concavity and its properties
・Understand the concavity maximum principle
・Understand the outline of recent development of power concavity properties of solutions to elliptic and parabolic equations
Keywords
power concavity, concavity maximum principle, log-concavity, parabolic power concavity, Minkowski addition
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | The following topics will be covered in this order : - power concavity and its properties - concavity maximum principle and its applications - non-preservation of quasi-concavity - preservation of log-concavity for parabolic equations - parabolic power concavity - parabolic equations and Minkowski addition - analysis of power concavity on manifolds | Details will be provided during each class session |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None required
Reference books, course materials, etc.
None
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.C341 ： Differential Equations I
- MTH.C342 ： Differential Equations II
Prerequisites
None