Neumann type boundary conditions for Hamilton-Jacobi equations

In this paper, we present a notion of viscosity solutions of Hamilton-Jacobi equations for Neumann type boundary conditions (or more generally oblique derivative). In particular we prove the existence, uniqueness, stability of such solutions and we show that the vanishing viscosity method yields such solutions. Next, we check that value functions of control problems or differential games problems for reflected dynamical processes are solutions in that sense of the associated Bellman or Isaacs equations. Finally, we consider the ergodic problems.