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We consider a general second order matrix operator in a multi-dimensionaldomain subject to a classical boundary condition. This operator is perturbed bya first order differential operator, the coefficients of which dependarbitrarily on a small multi-dimensional parameter. We study the existence of alimiting (homogenized) operator in the sense of the norm resolvent convergencefor such perturbed operator. The first part of our main results states that thenorm resolvent convergence is equivalent to the convergence of the coefficientsin the perturbing operator in certain space of multipliers. If this is thecase, the resolvent of the perturbed operator possesses a complete asymptoticexpansion, which converges uniformly to the resolvent. The second part of ourresults says that the convergence in the mentioned spaces of multipliers isequivalent to the convergence of certain local mean values over small pieces ofthe considered domains. These results are supported by series of examples. Wealso provide a series of ways of generating new non-periodically oscillatingperturbations, which finally leads to a very wide class of perturbations, forwhich our results are applicable.

Homogenization for operators with arbitrary perturbations in coefficients