Hamilton-Jacobi Equations: Viscosity Solutions, Optimal Control and Periodic Homogenization

The course is an introduction to the viscosity theory of first-order Hamilton-Jacobi equations with applications. A model example is the equation \begin{equation} \label{eq}\left\{\begin{array}{ll} u_t=H(\nabla u, x):=\lvert\nabla u\rvert^2+f(x) & \textrm{on}\;\;\mathbb{R}^d\times(0,\infty), \\ u=u_0 & \textrm{on}\;\;\mathbb{R}^d\times\left\{0\right\}.\end{array}\right. (1) \end{equation} The goals of the lecture series are fourfold:

• To explain the relationship between optimal control theory and solutions to convex Hamilton-Jacobi equations of the type (1).
• To show by explicit example and through an analysis of the associated characteristics that smooth solutions to equations like (1) do not exist in general, even for smooth Hamiltonians H.
• To present the viscosity formulation of equation (1), and to prove the fundamental results concerning the existence and uniqueness of viscosity solutions.
• To prove the periodic homogenization of equations like \begin{equation}\label{hom}\left\{\begin{array}{ll} u^\epsilon_t=H(\nabla u^\epsilon,\frac{x}{\epsilon}) & \textrm{on}\;\;\mathbb{R}^d\times(0,\infty), \\ u^\epsilon=u_0 & \textrm{on}\;\;\mathbb{R}^d\times\left\{0\right\},\end{array}\right.      (2)\end{equation} for Hamiltonians H which are periodic in space, and to show under general assumptions that the homogenization occurs with an algebraic rate.

Date and time info
Thursday 16:00 - 17:30

Keywords
First-Order Hamilton-Jacobi Equations, Viscosity Solutions, Optimal Control Theory, Periodic Homogenization

Prerequisites
Calculus, Measure Theory, Basic Functional Analysis

Audience
MSc students, PhD students, Postdocs

Language
English

Remarks and notes
The course is self-contained, and assumes no previous knowledge of the aforementioned topics.