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

TL;DR

This paper develops a method to analyze the homogenization of quasiperiodic monotone operators by transforming the problem into a higher-dimensional periodic setting, providing a new convergence characterization and corrector results.

Contribution

It introduces a two-scale cut-and-projection convergence framework for nonlinear monotone PDEs with quasiperiodic structures, linking them to periodic homogenization in higher dimensions.

Findings

Characterizes the limit of quasiperiodic monotone operators via cut-and-projection.

Identifies the homogenized problem as a local PDE on a hyperplane in higher-dimensional space.

Establishes a new corrector result for the homogenization process.

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 $-\mathrm{div} \; \sigma\left({\bf x},\frac{{\bf R}{\bf x}}{\eta}, \nabla u_\eta\right)$ for a bounded sequence $u_\eta$ in $W^{1,p}_0(\Omega)$, 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.