## Weak KAM Theorem: Lagrangian Dynamics and viscosity solutions of PDEs

Department:
MATH
Course Number:
8803-FAT
Hours - Lecture:
3
Hours - Lab:
0
Hours - Recitation:
0
Hours - Total Credit:
0
Typical Scheduling:
Not typically scheduled
Prerequisites:

Math 6337 (Real Analysis I) Math 6338 (Real Analysis II)  Math Ordinary Differential Equations I

Course Text:

• Weak KAM Theorem in Lagrangian Dynamics, Lecture notes, 10th version (2008) https://tinyurl.com/ycx8geeb
• -Sorrentino, Action-minimizing Methods in Hamiltonian Dynamics: An Introduction to Aubry-Mather Theory Princeton
• -Cannarsa  & Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Birkhäuser
• -Bardi & Italo Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations Birkhäuser
Topic Outline:

The subject is at the crossroad of Lagrangian dynamical systems (Aubry-Mather theory) and the viscosity solutions of the Hamilton-Jacobi equation (a first order PDE). Its historical roots are classical mechanical systems. It has also deep roots in optimization and what is now called tropical geometry.

The subject can be done with very little amount beyond the common knowledge of a  graduate student. Although the natural framework is manifolds, most of the theory can be done on Euclidean space and the torus. Therefore the framework will be adapted to the audience.

Topics covered are:

• -dual convex functions.
• -Classical calculus of variations.
• -Lagrangian and Hamiltonian dynamics
• -Hamilton-Jacobi equation and dynamics
• -Viscosity solutions of Hamilton-Jacobi equation and dynamics
• -Aubry  and Mather sets. Minimizing measures

Time permitting we will give some applications of the methods:

• -discrete weak KAM theory.
• -Lyapunov functions in dynamics
• -time functions in Lorentz spaces.
• -distance functions to closed sets in Euclidean space (or general Riemannian manifolds).