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 homogenized tensors.
Bloch Wave Homogenization of Quasiperiodic Media
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abstract
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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2019 1verdicts
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Bloch wave approach to almost periodic homogenization and approximations of effective coefficients
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 homogenized tensors.