On homogenization for non-self-adjoint locally periodic elliptic operators

Abstract

In this note we consider the homogenization problem for a matrix locally periodic elliptic operator on R^d of the form A^ε = −divA(x, x/ε)∇. The function A is assumed to be Hölder continuous with exponent s ∈ [0, 1] in the “slow” variable and bounded in the “fast” variable. We construct approximations for (A^ε − μ)⁻¹, including one with a corrector, and for (−Δ)^s/2(A^ε − μ)⁻¹ in the operator norm on L₂(R^d)ⁿ. For s ≠ 0, we also give estimates of the rates of approximation.