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- Title
Homogenization of an Elliptic System as the Cells of Periodicity are Refined in One Direction.
- Authors
Nazarov, S. A.; Slutskii, A. S.
- 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 $$\kappa$$ ∈ (0, 1/2] of smoothness of the right-hand side with respect to slow variables (the Sobolev-Slobodetskii space) and is estimated by $$O(h^\kappa )$$ for $$\kappa$$ ∈ (0, 1/2) and by O( h 1/2(1 + |log h|)) for $$\kappa$$ = 1/2.
- Subjects
ELLIPTIC space; NON-Euclidean geometry; ELLIPTIC surfaces; ASYMPTOTIC homogenization; PARTIAL differential equations; DIRICHLET problem; DIRICHLET forms
- Publication
Mathematical Notes, 2005, Vol 78, Issue 5/6, p814
- ISSN
0001-4346
- Publication type
Article
- DOI
10.1007/s11006-005-0187-8