Spectral Approach to Homogenization of Elliptic Operators in a Perforated Space

Abstract

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.