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

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...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Wellander, Niklas, Guenneau, Sebastien, Cherkaev, Elena
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 08.06.2022
Subjects
Convergence
Differential equations
Hyperplanes
Mathematics - Analysis of PDEs
Mathematics - Mathematical Physics
Operators (mathematics)
Periodic structures
Physics - Materials Science
Physics - Mathematical Physics
Projection
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2206.03672