Two-parametric error estimates in homogenization of second order elliptic systems in Rᵈ including lower order terms

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In L₂(Rᵈ;Cⁿ), we consider a selfadjoint operator Bε, 0< ε 1, given by the differential expression b(D)* g(x/ε)b(D) + ∑ⱼ₌₁ᵈ (aⱼ(x/ε) Dⱼ +Dⱼ aⱼ(x/ε)*) + Q(x/ε), where b(D) = ∑ₗ₌₁ᵈ bₗ Dₗ is the first order differential operator, and g, aⱼ, Q are matrix-valued functions in Rᵈ periodic with respect to some lattice Γ. It is assumed that g is bounded and positive definite, while aⱼ and Q are, in general, unbounded. We study the generalized resolvent (Bε - ζQ₀(x/ε))⁻¹, where Q₀ is a Γ-periodic, bounded and positive definite matrix-valued function, and ζ is a complex-valued parameter. Approximations for the generalized resolvent in the (L₂ → L₂)- and (L₂ → H¹)-norms with two-parametric error estimates (with respect to the parameters ε and ζ) are obtained.