2021 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on current topics in Mathematics R
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Hiroaki Aikawa / Yoshihiro Tonegawa
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive
- Class
- -
- Course Code
- MTH.E647
- Number of credits
- 200
- Course offered
- 2021
- Offered quarter
- 2Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
The main subject of this course is positive (super) solutions to elliptic equations and parabolic equations over non-smooth domains. There is a long history on solutions to the Laplace equation and the heat equations; their precise properties have been well-exploited. However, their behavior near the boundary still remains in mystery if the domain is non-smooth. This course focuses on the global integrability of positive superharmonic functions and positive supersolutions to the heat equation with explanation of basic properties of these supersolutions. Our main objective is non-smooth Euclidean domains. Non-smooth domains in a manifold may be touched if time allows.
We aim to deepen understanding of various notions and techniques in mathematical analysis through the study on the integrability of positive superharmonic functions and positive supersolutions to the heat equations.
Course description and aims
Be familiar with the following items:
-- Green function, heat kernel, Riesz theorem, capacity
-- Lipschitz domain, John domain, quasihyperbolic metric
-- Martin boundary, boundary Harnack principle, heat kernel estimate
-- Cranston-McConnell inequality, intrinsic ultracontractivity
-- harmonic measure, survival probability, capacitary width, box argument
Keywords
Elliptic equation, parabolic equation, supersolution, integrability, Green function, heat kernel
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is an intensive lecture course. Assignment will be provided during each class session.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | The global integrability of positive supersolutions to the heat equation is based on the following ingredients, which will be illustrated in detail through the lecture: -- Green function, heat kernel, Riesz theorem, capacity -- Lipschitz domain, John domain, quasihyperbolic metric -- Martin boundary, boundary Harnack principle, heat kernel estimate -- Cranston-McConnell inequality, intrinsic ultracontractivity -- harmonic measure, survival probability, capacitary width, box argument | 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.
Provided during each class session.
Evaluation methods and criteria
Assignments (100%).
Related courses
- none
Prerequisites
None required.