Homogenization of an elliptic system as the cells of periodicity are refined in one direction

S Nazarov, AS Slutskii

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Abstract

We homogenize a second-order elliptic system with anisotropic fractal structure characteristic of many real objects: the cells of periodicity are refined in one direction. This problem is considered in the rectangle with Dirichlet conditions given on two sides and periodicity conditions on two other sides. An explicit formula for the homogenized operator is established, and an asymptotic estimate of the remainder is obtained. The accuracy of approximation depends on the exponent ¿ ¿ (0, 1/2] of smoothness of the right-hand side with respect to slow variables (the Sobolev¿Slobodetskii space) and is estimated by O(h¿) for ¿ ¿ (0, 1/2) and by O(h1/2(1 + | log h|)) for ¿ = 1/2 . Key words: elliptic system of second order, anisotropic fractal structure, q-branching function, branching periodicity, Sobolev¿Slobodetskii space, Dirichlet condition.
Original languageUndefined/Unknown
Pages (from-to)814-826
Number of pages13
JournalAcademy of Sciences of the USSR. Mathematical Notes
Volume78
Issue number5-6
Publication statusPublished - 2005

Keywords

  • academic journal papers
  • CWTS JFIS < 0.75

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