TOKYO INSTITUTE OF TECHNOLOGY

Course description and aims

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.

Student learning outcomes

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 that will be developed

Class flow

This is an intensive lecture course. Assignment will be provided during each class session.

Course schedule/Required learning

Textbook(s)

None required

Reference books, course materials, etc.

Provided during each class session.

Assessment criteria and methods

Assignments (100%).

Related courses

• None

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

None Required