Some homogenization results for non-coercive Hamilton–Jacobi equations

Abstract

Recently, C. Imbert & R. Monneau study the homogenization of coercive HamiltonJacobi Equations with a u/ε-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/ε-dependence) but in the case of 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.

Homogenization of First-Order Equations with $$(u/\varepsilon)$$ -Periodic Hamiltonians. Part I: Local Equations

“…After this work was completed, Barles [4] provided much simpler proofs of our results in a special case. Moreover, the extensive use of these ideas permited him to obtain homogenization results for noncoercive Hamiltonians.…”

“…Equation (4). The second thing, is that Ansatz (7) is not the right Ansatz, because the final result depends on the term…”

Homogenization of fully overdamped Frenkel–Kontorova models

“…Let us mention the recent result of Imbert, Monneau [15] and the one of Barles [6]. We can also mention the work of Boccardo, Murat [7] about the homogenization of elliptic equations and the one of Bacaër [4].…”

Homogenization of monotone systems of Hamilton-Jacobi equations

“…When the Hamiltonians are not coercive anymore (and therefore the corrector is not necessarily Lipschitz continuous), the proof is more delicate. A way to solve this problem is to use the ideas of Barles [4] and his "F k -trick" (see [4], Lem. 2.1 and Thm.…”