Two-scale cut-and-projection convergence for quasiperiodic monotone operators

Abstract

Averaging a certain class of quasiperiodic monotone operators can be simplified to the periodic homogenization setting by mapping the original quasiperiodic structure onto a periodic structure in a higher dimensional space using the cut-and projection method. We characterize the cut-and-projection convergence limit of the nonlinear monotone partial differential operator −divσx,Rxη,∇uη for a bounded sequence uη in W01,p(Ω), where 1<p<∞, and Ω is a bounded open subset in Rn with Lipschitz boundary. We identify the homogenized problem with a local equation defined on a hyperplane, or a lower dimensional plane in the higher-dimensional space. A new corrector result is established.