A Variational Approach to the Stability in the Homogenization of Some Hamilton–Jacobi Equations

Abstract. We investigate the stability with respect to homogenization of classes of integrals arising in the control-theoretic interpretation of some Hamilton–Jacobi equations. The prototypical case is the homogenization of energies with a Lagrangian consisting of the sum of a kinetic term and a highly oscillatory potential [Formula: see text], where [Formula: see text] is periodic and [Formula: see text] is a nonnegative perturbation thereof. We assume that [Formula: see text] has zero average in tubular domains oriented along a dense set of directions. Stability then holds true; that is, the resulting homogenized functional is identical to that for [Formula: see text]. We consider various extensions of this case. As a consequence of our results, we obtain stability for the homogenization of some steady-state and time-dependent, first-order Hamilton–Jacobi equations with convex Hamiltonians and perturbed periodic potentials. Finally, we show with an example that, for negative [Formula: see text], stability may not hold. Our study revisits and, depending on the different assumptions, complements results obtained by P.-L. Lions, P. Souganidis, and their collaborators using PDE techniques.