Special lectures on current topics in Mathematics L

Academic unit or major

Graduate major in Mathematics

Instructor(s)

Hiroyoshi Mitake / Yoshihiro Tonegawa

Class Format

Lecture (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

Intensive (本館2階201数学系セミナー室)

Class

-

Course Code

MTH.E642

Number of credits

200

Course offered

2023

Offered quarter

4Q

Syllabus updated

Jul 8, 2025

Language

Japanese

Syllabus

Course overview and goals

The main subject of this course is the theory of viscosity solutions which is one of tools to study nonlinear partial differential equations which are classified as elliptic and parabolic equations. The theory of viscosity solutions was introduced by Crandall and Lions early 1980s, and is a standard tool to study fully nonlinear partial differential equations, for instance, Hamilton-Jacobi equations, and some of level set equations of geometric flow, representatively, mean curvature flow.
In the course, we learn fundamental results in the theory of viscosity solutions including the comparison principle, existence, stability, representation formula for solutions etc. Moreover, we study a level-set forced mean curvature flow with the homogeneous Neumann boundary condition. In particular, we learn a way to obtain the time global Lipschitz estimate of viscosity solutions, and the large-time asymptotics, which are rather recent results in this field.

Course description and aims

(a) Understand the definition of viscosity solutions
(b) Understand the proofs for fundamental results (comparison principle, existence, stability for first order equations) in the theory of viscosity solutions
(c) Understand Crandall-Lions lemma, and be able to apply to the comparison principle for second order equations
(d) Understand the representation formula for viscosity solutions
(e) Understand Lipschitz regularity estimate
(f) Understand the time global Lipschitz estimate of viscosity solutions to a level-set forced mean curvature flow with the homogeneous Neumann boundary condition, and the large-time asymptotics.

Keywords

Viscosity solution, Level set approach, Geometric flow, Hamilton-Jacobi equation, Mean curvature flow

Competencies

• 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/Objectives

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.

Yoshikazu Gig, Surface Evolution Equations, Birkhauser (2006)，
Hung Vinh Tran，Hamilton-Jacobi Equations: Theory and Application, American Mathematical Society (2021)

Evaluation methods and criteria

Assignments (100%). Attendance to each class sessions is required too.

Related courses

• MTH.C351 ： Functional Analysis

Prerequisites

None