Journal
ST PETERSBURG MATHEMATICAL JOURNAL
Volume 29, Issue 2, Pages 325-362Publisher
AMER MATHEMATICAL SOC
DOI: 10.1090/spmj/1496
Keywords
Periodic differential operators; higher-order elliptic equations; Dirichlet problem; homogenization; effective operator; corrector; operator error estimates
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Funding
- RFBR [16-01-00087]
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Let O subset of R-d be a bounded domain of class C-2p. The object under study is a selfadjoint strongly elliptic operator A(D,epsilon) of order 2p, p >= 2, in L-2(O;C-n), given by the expression b(D)* g(x/epsilon)b(D), epsilon > 0, with the Dirichlet boundary conditions. Here g(x) is a bounded and positive definite (m x m)-matrix-valued function in R-d, periodic with respect to some lattice; b(D) = Sigma(|alpha|=p) b(alpha)D(alpha) is a differential operator of order p with constant coefficients; and the b(alpha) are constant (m x n)-matrices. It is assumed that m >= n and the symbol b(xi) has maximal rank. Approximations are found for the resolvent (Lambda(D,epsilon) -zeta I)(-1) in the L-2(O;C-n)-operator norm and in the norm of operators acting from L-2(O;C-n) to H-p(O;C-n), with error estimates depending on epsilon and zeta.
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