2025 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Analysis B1
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Michiaki Onodera
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Mon (M-102(H115))
- Class
- -
- Course Code
- MTH.C406
- Number of credits
- 100
- Course offered
- 2025
- Offered quarter
- 2Q
- Syllabus updated
- Mar 19, 2025
- Language
- English
Syllabus
Course overview and goals
This lecture and "Advanced topics in Analysis A1" are aimed at learning several aspects of overdetermined boundary value problems for elliptic partial differential equations, with a special emphasis on its analytical methods.
Here, an overdetermined boundary value problem refers to a boundary value problem on which excessive boundary conditions are imposed; so that our interet lies in a special domain (or its boundary known as "free boundary") that allows such a problem to have a solution.
A typical example is a second-order elliptic equation with both Dichlet and Neumann boundary conditions.
Such a problem arises as a free boundary value problem in fluid dynamics and the Euler-Lagrange equation of a shape optimization problem.
In this lecture, we study basic properties of such a special domain, including existence, uniqueness and stability under some perturbation of boundary conditions.
A particular emphasis is placed on its methods such as the method of moving planes, variational method and several perturbative approaches.
Course description and aims
Understanding the methods and ideas in partial differential equations involving domain variations.
Keywords
Partial differential equations, Overdetermined problems, Maximum principles, Variational methods, Analytic semigroups
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course. Occasionally I will give problems for reports.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Variational structures of overdetermined problems 1 | Details will be provided in the class. |
| Class 2 | Variational structures of overdetermined problems 1 | Details will be provided in the class. |
| Class 3 | One-parameter family of overdetermined problems and geometric flow 1 | Details will be provided in the class. |
| Class 4 | One-parameter family of overdetermined problems and geometric flow 2 | Details will be provided in the class. |
| Class 5 | Analytic semigroups | Details will be provided in the class. |
| Class 6 | Theory of maximal regularity | Details will be provided in the class. |
| Class 7 | Solvability of geometric flow | Details will be provided in the class. |
Study advice (preparation and review)
Sufficiently prepare for and review each class.
Textbook(s)
Not required
Reference books, course materials, etc.
- D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001.
- A. Henrot, M. Pierre, Shape variation and optimization, European Mathematical Society, 2018.
- A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser, 1995.
Evaluation methods and criteria
Report (100%)
Related courses
- MTH.C405 ： Advanced topics in Analysis A1
Prerequisites
Students are required to take Advanced topics in Analysis A1 (MTH.C405).