Beijing Institute of Mathematical Sciences and Applications

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.