Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients
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Let O ⊂ R d be a bounded domain of class C 1,1. Let 0 < ε - 1. In L 2(O;C n ) 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 − ζQ 0(·/ε))−1 as ε → 0. Here the matrix-valued function Q 0 is periodic, bounded, and positive definite; ζ is a complex-valued parameter. We find approximations of the generalized resolvent in the L 2(O;C n )-operator norm and in the norm of operators acting from L 2(O;C n ) to the Sobolev space H 1(O;C n ) 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 Q 0(x/ε)∂ t v ε (x, t) = −(B D,ε v ε )(x, t).
Key wordsperiodic differential operators elliptic systems parabolic systems homogenization operator error estimates
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