Calculus of Variations and Geometric Measure Theory

A. Braides - G. Dal Maso - C. Le Bris

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

created by braidesa on 29 Oct 2024
modified on 01 Nov 2024

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Preprint

Inserted: 29 oct 2024
Last Updated: 1 nov 2024

Year: 2024

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.

Keywords: Homogenization, stability, Hamilton-Jacobi equations, perturbation


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