Bloch wave approach to almost periodic homogenization

Abstract: Bloch wave homogenization is a spectral method for obtaining effective coefficients for periodically heterogeneous media. This method hinges on the direct integral decomposition of periodic operators, which is not available in a suitable form for almost periodic operators. In particular, the notion of Bloch eigenvalues and eigenvectors does not exist for almost periodic operators. However, we are able to recover the homogenization result in this case, by employing a sequence of periodic approximations to almost periodic operators. Another approach, that employs periodic lifting of quasiperiodic operators is also discussed. We also establish a rate of convergence for approximations of homogenized tensors for a class of almost periodic media.