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

Journal article (2005)

Authors

S Nazarov Analysis - EEMCS
AS Slutskii External organisation
Research Group
Analysis (EEMCS) (TU Delft)
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Published Date

2005

Language

Undefined/Unknown

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

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.