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

Abstract

Averaging 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 cut-and projection method. We characterize cut-and-projection convergence limit of the nonlinear monotone partial differential operator $-\mathrm{div}\sigma \left(\mathbf{x},\frac{\mathbf{R}\mathbf{x}}{\eta },\nabla u_{\eta }\right)$ for a bounded sequence $u_{\eta }$ in $W_0^{1,p}(\Omega )$, where $1<p<\infty$, $\Omega$ is a bounded open subset in $R^n$ with Lipschitz boundary. We identify the homogenized problem with a local equation defined on the hyperplane in the higher-dimensional space. A new corrector result is established.

18 pages