Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations

The purpose of this paper is to prove long-time behaviour results for Hamilton-Jacobi equations. For autonomous equations, we give an alternative proof of a convergence theorem obtained by A. Fathi when the equations are posed on a manifold, then extend it to Dirichlet boundary conditions on an open subset. When the equations are time-periodic we prove the convergence in several nontrivial special cases. (C) 2001 Editions scientifiques et medicales Elsevier SAS.