2026 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Analysis B
- 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 Thu (W5-104)
- Class
- -
- Course Code
- MTH.C402
- Number of credits
- 100
- Course offered
- 2026
- Offered quarter
- 2Q
- Syllabus updated
- Mar 5, 2026
- Language
- English
Syllabus
Course overview and goals
This course provides an exposition of the fundamental theory of second-order elliptic partial differential equations.
In particular, we will discuss in detail the theory of weak solutions based on Sobolev spaces, and examine the existence and regularity of solutions using functional analysis and variational methods.
Furthermore, we will cover the De Giorgi-Nash-Moser theory, which is an essential tool for investigating the regularity of solutions to nonlinear equations.
This course is designed as a continuation of "Advanced Topics in Analysis A".
Course description and aims
Understanding of the basic theory of overdetermined problems for elliptic partial differential equations
Keywords
elliptic partial differential equations, Sobolev spaces, calculus of variations, regularity theory
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 | Sobolev spaces |
Details will be provided during each class session. |
| Class 2 | Dirichlet problem |
|
| Class 3 | Calculus of variations |
|
| Class 4 | Regularity theory 1 |
|
| Class 5 | Regularity theory 2 |
|
| Class 6 | Regularity theory 3 |
|
| Class 7 | Advanced topics |
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
Reference books, course materials, etc.
D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001.
Evaluation methods and criteria
Report (100%)
Related courses
- MTH.C305 ： Real Analysis I
- MTH.C306 ： Real Analysis II
- MTH.C351 ： Functional Analysis
- MTH.C401 ： Advanced topics in Analysis A
Prerequisites
None
Other
None