Résumé
In this paper, we present a result of homogenization of first order Hamilton-Jacobi equations with ($u/\varepsilon$)-periodic Hamiltonians. On the one hand, under a coercivity assumption on the Hamiltonian (and some natural regularity assumptions), we prove an ergodicity property of this equation and the existence of non periodic approximate correctors. On the other hand, the proof of the convergence of the solution, usually based on the introduction of a perturbed test function in the spirit of Evans' work, uses here a twisted perturbed test function for a higher dimensional problem.