A variational approach to the stability in the homogenization of some Hamilton-Jacobi equations

TL;DR

This paper studies the stability of homogenization for certain Hamilton-Jacobi equations with oscillatory potentials, showing stability under specific conditions and providing counterexamples for negative perturbations.

Contribution

It introduces a variational approach to analyze stability in homogenization of Hamilton-Jacobi equations with oscillatory potentials, extending previous PDE-based results.

Findings

Stability holds when the perturbation W has zero average in certain tubular domains.

Homogenized functional remains unchanged for specific oscillatory perturbations.

Counterexample shows stability may fail for negative perturbations W.

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 $V =V_{\rm per}+ W$, where $V_{\rm per}$ is periodic and $W$ is a nonnegative perturbation thereof. We assume that $W$ 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 $W= 0$. 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 $W$, stability may not hold. Our study revisits and, depending on the different assumptions, complements results obtained by P.-L. Lions and collaborators using PDE techniques.