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- Academic unit or major
- Mathematics
- Instructor(s)
- Tonegawa Yoshihiro
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- Day/Period(Room No.)
- Intensive (本館2階201数学系セミナー室)
- Group
- -
- Course number
- ZUA.E346
- Credits
- 2
- Academic year
- 2023
- Offered quarter
- 4Q
- Syllabus updated
- 2023/3/20
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
-
Course description and aims
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.
Student learning outcomes
(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 that will be developed
| ✔ 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/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | The following topics will be covered in this order : (a) Definition of viscosity solutions (b) Fundamental results (comparison principle, existence, stability for first order equations) in the theory of viscosity solutions (c) Crandall-Lions lemma, and the comparison principle for second order equations (d) Representation formula for viscosity solutions (e) Lipschitz regularity estimate (f) 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. | Details will be provided during each class session. |
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)
Assessment criteria and methods
Assignments (100%). Attendance to each class sessions is required too.
Related courses
- MTH.C341 ： Differential Equations I
- MTH.C342 ： Differential Equations II
Prerequisites (i.e., required knowledge, skills, courses, etc.)
None
