TY - JOUR

T1 - Homogenization of an elliptic system as the cells of periodicity are refined in one direction

AU - Nazarov, S. A.

AU - Slutskii, A. S.

PY - 2005/11/1

Y1 - 2005/11/1

N2 - 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 κ (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 x ) for x ∈ (0, 1/2) and by O(h 1/2(1 + |log h|)) for x = 1/2.

AB - 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 κ (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 x ) for x ∈ (0, 1/2) and by O(h 1/2(1 + |log h|)) for x = 1/2.

KW - Anisotropic fractal structure

KW - Branching periodicity

KW - Dirichlet condition

KW - Elliptic system of second order

KW - Q-branching function

KW - Sobolev-Slobodetskii space

UR - http://www.scopus.com/inward/record.url?scp=28644439092&partnerID=8YFLogxK

U2 - 10.1007/s11006-005-0187-8

DO - 10.1007/s11006-005-0187-8

M3 - Article

AN - SCOPUS:28644439092

VL - 78

SP - 814

EP - 826

JO - Mathematical Notes

JF - Mathematical Notes

SN - 0001-4346

IS - 5-6

ER -