Homogenization of high order elliptic operators with periodic coefficients

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In L₂(Rᵈ;Cⁿ), we study a selfadjoint strongly elliptic operator Aε of order 2p given by the expression b(D)* g(x/ε) b(D), ε >0. Here g(x) is a bounded and positive definite (m× m)-matrix-valued function in Rᵈ; it is assumed that g(x) is periodic with respect to some lattice. Next, b(D)=∑|α|₌ₚᵈ bαDα is a differential operator of order p with constant coefficients; bα are constant (m× n)-matrices. It is assumed that m≥ n and that the symbol b(ξ) has maximal rank. For the resolvent (Aε - ζI)⁻¹ with ζ∈ C [0,∞), we obtain approximations in the norm of operators in L₂(Rᵈ;Cⁿ) and in the norm of operators acting from L₂(Rᵈ;Cⁿ) to the Sobolev space Hᵖ(Rᵈ;Cⁿ), with error estimates depending on ε and ζ.