Some Homogenization Results for Non-Coercive Hamilton-Jacobi Equations

Recently, C. Imbert & R. Monneau study the homogenization of coercive Hamilton-Jacobi Equations with a $u\mathrm{/}e$-dependence : this unusual dependence leads to a non-standard cell problem and, in order to solve it, they introduce new ideas to obtain the estimates on the oscillations of the solutions. In this article, we use their ideas to provide new homogenization results for ``standard’’ Hamilton-Jacobi Equations (i.e. without a $u\mathrm{/}e$-dependence) but in the case of {\it non-coercive Hamiltonians}. As a by-product, we obtain a simpler and more natural proof of the results of C. Imbert & R. Monneau, but under slightly more restrictive assumptions on the Hamiltonians.