Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients

Abstract

Let O ⊂ R^d be a bounded domain of class C^1,1. Let 0 < ε - 1. In L₂(O;Cⁿ) we consider a positive definite strongly elliptic second-order operator B_D,ε with Dirichlet boundary condition. Its coefficients are periodic and depend on x/ε. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent (B_D,ε − ζQ₀(·/ε))⁻¹ as ε → 0. Here the matrix-valued function Q₀ is periodic, bounded, and positive definite; ζ is a complex-valued parameter. We find approximations of 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-parameter error estimates (depending on ε and ζ). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation Q₀(x/ε)∂_tv_ε(x, t) = −(B_D,εv_ε)(x, t).