These Lecture Notes contain the material relative to the courses given at the CIME summer
school held in Cetraro Italy from August 29 to September 3 2011. The topic was
Hamilton-Jacobi Equations: Approximations Numerical Analysis and Applications. The courses
dealt mostly with the following subjects: first order and second order Hamilton-Jacobi-Bellman
equations properties of viscosity solutions asymptotic behaviors mean field games
approximation and numerical methods idempotent analysis. The content of the courses ranged
from an introduction to viscosity solutions to quite advanced topics at the cutting edge of
research in the field. We believe that they opened perspectives on new and delicate issues.
These lecture notes contain four contributions by Yves Achdou (Finite Difference Methods for
Mean Field Games) Guy Barles (An Introduction to the Theory of Viscosity Solutions for
First-order Hamilton-Jacobi Equations and Applications) Hitoshi Ishii (A Short Introduction to
Viscosity Solutions and the Large Time Behavior of Solutions of Hamilton-Jacobi Equations) and
Grigory Litvinov (Idempotent Tropical Analysis the Hamilton-Jacobi and Bellman Equations).
