2021 Special lectures on advanced topics in Mathematics L

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Ishige Kazuhiro  Onodera Michiaki 
Course component(s)
Lecture    (ZOOM)
Day/Period(Room No.)
Intensive ()  
Group
-
Course number
MTH.E536
Credits
2
Academic year
2021
Offered quarter
4Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
Japanese
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Course description and aims

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.

Student learning outcomes

・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 that will be developed

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/Required learning

  Course schedule Required learning
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.

Textbook(s)

None required

Reference books, course materials, etc.

None

Assessment criteria and methods

Assignments (100%).

Related courses

  • MTH.C341 : Differential Equations I
  • MTH.C342 : Differential Equations II

Prerequisites (i.e., required knowledge, skills, courses, etc.)

None

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