Bloch Wave Homogenization of Quasiperiodic Media

Sivaji Ganesh Sista; Vivek Tewary

Bloch Wave Homogenization of Quasiperiodic Media

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arxiv 1910.12724 v1 pith:2I4EWIAL submitted 2019-10-28 math.AP

Sivaji Ganesh Sista , Vivek Tewary This is my paper · ORCID

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keywords periodicblochmediaquasiperiodicalmostequationcoefficientsdecomposition

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Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a "cut and project" procedure. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift an almost periodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularized version of the degenerate equation. Homogenized coefficients for quasiperiodic media are determined in terms of the first Bloch eigenvalue of the regularized lifted equation. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenization limit for an equation with highly oscillating quasiperiodic coefficients.

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Bloch wave approach to almost periodic homogenization and approximations of effective coefficients
math.AP 2019-08 conditional novelty 5.0 of 10

Using periodic approximations and the first Bloch eigenvalue, the authors recover the homogenization limit for almost periodic elliptic operators and prove a power-law rate for the convergence of approximate homogeniz...

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