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2021, Volume 16, Issue 3: 427-458. Doi: 10.3934/nhm.2021012

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Combined effects of homogenization and singular perturbations: A bloch wave approach

Vivek Tewary

Tata Institute of Fundamental Research, Centre for Applicable Mathematics, Bangalore, Karnataka, 560065, India

Received: December 2020

Early access: May 2021

Published: September 2021

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Abstract

In this work, we study Bloch wave homogenization of periodically heterogeneous media with fourth order singular perturbations. We recover different homogenization regimes depending on the relative strength of the singular perturbation and length scale of the periodic heterogeneity. The homogenized tensor is obtained in terms of the first Bloch eigenvalue. The higher Bloch modes do not contribute to the homogenization limit. The main difficulty is the presence of two parameters which requires us to obtain uniform bounds on the Bloch spectral data in various regimes of the parameter.

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Mathematics Subject Classification: 35B27, 74Q05, 47A55, 34E15, 35P05.

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