Linear elliptic homogenization for a class of highly oscillating non-periodic potentials

Abstract

We consider an homogenization problem for the second order elliptic equation −∆u ε + 1 ε V (./ε)u ε + νu ε = f when the highly oscillatory potential V belongs to a particular class of non-periodic potentials. We show the existence of an adapted corrector and prove the convergence of u ε to its homogenized limit.

Defects in homogenization theory

“…In the absence of a general strategy for solving this question, we have only been able to consider some specific instances of this general problem. The most recent example in this line of research is the work [33] where homogenization of the Schrödinger equation −∆u ε + ε −α V (./ε) u ε = f is considered for a general class of highly oscillatory potentials V constructed using a set of points X k as above.…”