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

Abstract

To cite this version:Cyril Imbert, Régis Monneau. Homogenization of first order equations with u/ϵ-periodic Hamiltonians. Abstract. In this paper, we present a result of homogenization of first order Hamilton-Jacobi equations with (u/ε)-periodic Hamiltonians. On the one hand, under a coercivity assumption on the Hamiltonian (and some natural regularity assumptions), we prove an ergodicity property of this equation and the existence of non periodic approximate correctors. On the other hand, the proof of the convergence of the solution, usually based on the introduction of a perturbed test function in the spirit of Evans' work, uses here a twisted perturbed test function for a higher dimensional problem.

Some homogenization results for non-coercive Hamilton–Jacobi equations

“…In fact, with our approach, we obtain a weaker but sufficient result which can take two possible forms : we can prove that there exists a unique constant λ such that we have either approximate continuous (but not necessarily Lipschitz continuous) periodic solutions or exact, possibly discontinuous, periodic sub and supersolutions. This result is the first main contribution of the paper and this is where we use in a key way the ideas of Imbert & Monneau [20] ; an unusual feature of the proof is the estimate of the oscillation max R n+1 ×R (w) − min R n+1 ×R (w) which replaces the classical gradient estimate and relies on two very original arguments. Once (4) is solved, the result for the homogenization problem follows by using the usual arguments : indeed, it is enough to (essentially) apply the result to (3) for any (p x , p y ), then to obtain suitable properties for F and finally to use the "perturbed test function's method" of Evans [18,19], even if, to prove the convergence, we have to introduce an additional argument to take care of the rather weak properties we impose on F in the variables x, t and p x .…”

“…The originality of this work is to provide results in the case of noncoercive Hamiltonians and applications to non-standard problems. Before describing more specifically our contributions, we want to point out that most of the new ideas used in this paper are borrowed from Imbert & Monneau [20] who study the homogenization of (coercive) Hamilton-Jacobi Equations with a u ε /ε-dependence, namely…”

“…The rigourous justification of this formal (but convincing) argument is easy by following the article of Giga & Sato [21] which is exactly describing the way to do it for firstorder equations (see also the paper of Biton, Ley & the author [14] for motion by Mean Curvature). Compared to the article of Imbert & Monneau [20], this approach leads to more restrictive assumptions on H but to a far simpler proof and to more natural and understandable arguments. Typically the results of [20] apply for all H of the form…”

“…Finally, it is worth pointing out that Imbert & Monneau [20] use also an extension to R n+1 by adding a extra variable but with no clear interpretation of this new variable which is just a trick in the proof.…”

“…By the arguments of Soner, Souganidis and the author [16], since u(x, t)−y is the solution of the level set equation (20) associated to the initial data u 0 (x) − y, we have χ(X, t) ≥ 1 1 {u(x,t)−y>0} and χ(X, t) ≤ 1 1 {u(x,t)−y≥0} in R n+1 × (0, +∞).…”