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

TL;DR

This paper studies the homogenization of a second-order elliptic equation with highly oscillatory, non-periodic potentials, establishing convergence to a homogenized limit and constructing an appropriate corrector.

Contribution

It introduces a method to handle non-periodic, highly oscillating potentials in elliptic homogenization and proves convergence results for the solutions.

Findings

Existence of an adapted corrector for the non-periodic potential case.

Convergence of solutions to a homogenized limit.

Extension of homogenization theory to a new class of non-periodic potentials.

Abstract

We consider an homogenization problem for the second order elliptic equation $- \Delta u^{\varepsilon} + \dfrac{1}{\varepsilon} V(./\varepsilon) u^{\varepsilon} + \nu u^{\varepsilon} =f$ 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 $u^{\varepsilon}$ to its homogenized limit.