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\(L^2\)-estimates for homogenization of elliptic operators. (English. Russian original) Zbl 1441.35030

J. Math. Sci., New York 244, No. 4, 671-685 (2020); translation from Probl. Mat. Anal. 101, 117-129 (2019).
Summary: We study homogenization of second order elliptic differential operators \(A_\epsilon\) whose coefficients are \(\epsilon \)-periodic and rapidly oscillate as \(\epsilon \rightarrow 0\). For the difference between the resolvent \((A_\epsilon + 1)^{-1}\) and its approximations we prove the operator estimate of order \(\epsilon^2\) in the operator \(L^2\)-norm by using the first approximation method and the Steklov smoothing operator.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B45 A priori estimates in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
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References:

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