Homogenization of the Dirichlet problem for elliptic systems: Two-parametric error estimates

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Let Oᵈ be a bounded domain of class C¹,¹. In L₂(O;Cⁿ), we study a selfadjoint matrix elliptic second order differential operator BD,ε, 0<ε 1, with the Dirichlet boundary condition. The principal part of the operator is given in a factorized form. The operator involves lower order terms with unbounded coefficients. The coefficients of BD,ε are periodic and depend on x/ε. We study the generalized resolvent (BD,ε-ζQ₀(·/ε))⁻¹, where Q₀ is a periodic bounded and positive definite matrix-valued function, and ζ is a complex-valued parameter. We obtain approximations for the generalized resolvent in the L₂(O;Cⁿ)-operator norm and in the norm of operators acting from L₂(O;Cⁿ) to the Sobolev space H¹(O;Cⁿ), with two-parametric error estimates (depending on ε and ζ).