An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications

Abstract

In this course, we first present an elementary introduction to the concept of viscosity solutions for first-order Hamilton-Jacobi Equations: definition, stability and comparison results (in the continuous and discontinuous frameworks), boundary conditions in the viscosity sense, Perron's method, Barron-Jensen solutions. etc. We use a running example on exit time control problems to illustrate the different notions and results. In a second part, we consider the large time behavior of periodic solutions of Hamilton-Jacobi Equations: we describe recents results obtained by using partial differential equations type arguments. This part is complementary of the course of H. Ishii which presents the dynamical system approach ("weak KAM approach"). © 2013 Springer-Verlag Berlin Heidelberg.