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

  • Lecturer: Benjamin Fehrman
  • Date: Thursday 16:00 - 17:30
  • Room: MPI MiS, A3 02
  • Language: English
  • Target audience: MSc students, PhD students, Postdocs
  • Keywords: First-Order Hamilton-Jacobi Equations, Viscosity Solutions, Optimal Control Theory, Periodic Homogenization
  • Prerequisites: Calculus, Measure Theory, Basic Functional Analysis
  • Remarks: The course is self-contained, and assumes no previous knowledge of the aforementioned topics.

Abstract

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.

Regular Lectures (Winter 2016/2017)

15.10.2018, 13:54