Homogenization for operators with arbitrary perturbations in coefficients

TL;DR

This paper establishes conditions under which a second order matrix operator with arbitrary small perturbations in coefficients converges to a homogenized limit, providing asymptotic expansions and broad applicability.

Contribution

It introduces a novel criterion linking coefficient convergence in multiplier spaces to resolvent convergence for operators with arbitrary perturbations.

Findings

Norm resolvent convergence is equivalent to coefficient convergence in multiplier spaces.

The resolvent admits a uniform asymptotic expansion under these conditions.

The results apply to a wide class of non-periodic oscillating perturbations.

Abstract

We consider a general second order matrix operator in a multi-dimensional domain subject to a classical boundary condition. This operator is perturbed by a first order differential operator, the coefficients of which depend arbitrarily on a small multi-dimensional parameter. We study the existence of a limiting (homogenized) operator in the sense of the norm resolvent convergence for such perturbed operator. The first part of our main results states that the norm resolvent convergence is equivalent to the convergence of the coefficients in the perturbing operator in certain space of multipliers. If this is the case, the resolvent of the perturbed operator possesses a complete asymptotic expansion, which converges uniformly to the resolvent. The second part of our results says that the convergence in the mentioned spaces of multipliers is equivalent to the convergence of certain local mean values over small pieces of the considered domains. These results are supported by series of examples. We also provide a series of ways of generating new non-periodically oscillating perturbations, which finally leads to a very wide class of perturbations, for which our results are applicable.