Theory of Viscosity Solutions: with Applications to Hamilton-Jacobi, Bellman, Isaacs, and Monge-Ampère PDEs
This class is an introduction to the theory of viscosity solutions, a particular type of weak solutions introduced by Pierre-Louis Lions and Michael Crandall in the 1980's, of some elliptic and parabolic PDE, including the Eikonal equation, Hamilton-Jacobi equation, Isaacs equation, Hamilton-Jacobi-Bellman equation, Monge-Ampere equation, among others. The class will begin with a general discussion about the reasoning and examples of weak solutions of certain PDEs, then cover the vanishing viscosity method and discuss classical sub- and super-solutions. Then, we will fully jump into the theory of viscosity solutions for the Dirichlet problem, learning about the comparison principles (uniqueness of viscosity solutions), Perron's method (for proving existence of viscosity solution), limit theorems of viscosity solutions, and finish with a discussion of more general boundary value conditions.
Lecturer
Date
4th March ~ 27th May, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday,Friday | 15:20 - 16:55 | A14-203 | ZOOM 12 | 815 762 8413 | BIMSA |
Prerequisite
Calculus, linear algebra, basic analysis, measure theory, very basic PDE theory
Syllabus
Week 1: Types of PDEs, parabolic, elliptic, linear, semi-linear, quasilinear, fully nonlinear
Week 2: Weak solutions of PDEs, example with Eikonal equation
Week 3: Introduction to Hamilton-Jacobi Equations and weak solutions, Vanishing viscosity method
Week 4: Classical sub- and super-solutions
Week 5-6: Definition of viscosity solutions
Week 7-8: Comparison Principles and Uniqueness of Viscosity Solutions
Week 9-10: Perron's Method and Existence of Viscosity Solutions
Week 11: Limits of Viscosity Solutions
Week 12: General Boundary Conditions
Week 2: Weak solutions of PDEs, example with Eikonal equation
Week 3: Introduction to Hamilton-Jacobi Equations and weak solutions, Vanishing viscosity method
Week 4: Classical sub- and super-solutions
Week 5-6: Definition of viscosity solutions
Week 7-8: Comparison Principles and Uniqueness of Viscosity Solutions
Week 9-10: Perron's Method and Existence of Viscosity Solutions
Week 11: Limits of Viscosity Solutions
Week 12: General Boundary Conditions
Reference
"Partial Differential Equations" by Lawrence Evans, "A Beginner's Guide to the Theory of Viscosity Solutions" by Shigeaki Koike, "User's Guide to Viscosity Solutions of Second Order Partial Differential Equations" by Michael Crandall, Hitoshi Ishii, and Pierre-Louis Lions, "Elliptic Partial Differential Equations of Second Order" by David Gilbarg and Neil Trudinger
Audience
Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
My research mostly consists of using tools of analysis and numerical analysis to investigate and compute solutions of problems in optimal transport with “unusual” cost functions. Applications of the mathematical work include optics inverse problems, computational mesh generation, sampling, and optimal control. I completed my Ph.D. thesis on numerical methods for fully nonlinear elliptic PDEs arising in optimal transport in 2022 working under Brittany Hamfeldt at the New Jersey Institute of Technology. From 2022 to 2025 I worked as a postdoc at the University of Texas at Austin under the supervision of Richard Tsai. I joined BIMSA in late May, 2025.