Bloch wave homogenisation of quasiperiodic media

Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a ‘cut and project’ procedure. Quasiperiodic media displays some extraordinary optical, electronic and conductivity properties which call for the development of methods to analyse their microstructures and effective behaviour. In this paper, we develop the method of Bloch wave homogenisation for quasiperiodic media. 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 a quasiperiodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularised version of the degenerate operator. Homogenised coefficients for quasiperiodic media are obtained from the first Bloch eigenvalue of the regularised operator in the limit of regularisation parameter going to zero. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenisation limit for an equation with highly oscillating quasiperiodic coefficients.

Spectral Approach to Homogenization of Elliptic Operators in a Perforated Space

Let [Formula: see text] be a lattice. For [Formula: see text], we consider the perforated space [Formula: see text] which is an [Formula: see text]-periodic open connected set with Lipschitz boundary. In [Formula: see text], we consider a self-adjoint strongly elliptic second-order differential operator [Formula: see text] with periodic coefficients depending on [Formula: see text]. We study the behavior of the resolvent [Formula: see text] for small [Formula: see text]. Approximations for this resolvent in the [Formula: see text] and [Formula: see text]-operator norms with sharp order error estimates are found. The results are obtained by the operator-theoretic (spectral) approach. General results are applied to particular periodic operators of mathematical physics: the acoustics operator, the elasticity operator, and the Schrödinger operator with a singular potential.