Special courses on advanced topics in Mathematics L

Academic unit or major

Mathematics

Instructor(s)

Kazuhiro Ishige / Michiaki Onodera

Class Format

Lecture

Media-enhanced courses

-

Day of week/Period
(Classrooms)

Intensive

Class

-

Course Code

ZUA.E346

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

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