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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Homogenization of an elliptic system under condensing perforation of the domain


Authors: S. A. Nazarov and A. S. Slutskii
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 17 (2005), nomer 6.
Journal: St. Petersburg Math. J. 17 (2006), 989-1014
MSC (2000): Primary 35J99
Published electronically: September 20, 2006
MathSciNet review: 2202047
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Abstract: Homogenization of a system of second-order differential equations is performed in the case of a nonuniformly perforated rectangle where the sizes of the holes and the distances between pairs of them decrease as the distance from one of the bases of the rectangle increases. The Neumann conditions are assumed on the boundaries of the holes. The formal asymptotics of the solution is constructed, which involves the usual Ansatz of homogenization theory and also some Ansätze typical of solutions of boundary-value problems in thin domains, in particular, exponential boundary layers. Justification of the asymptotics is done with the help of the Korn inequality, which is proved for the perforated domain $ \Omega(h)$. Depending on the properties of the right-hand side, the norm of the difference between the true and the approximate solutions in the Sobolev space $ H^1(\Omega(h))$ is estimated by the quantity $ ch^\varkappa$ with $ \varkappa\in(0,1/2]$.


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Additional Information

S. A. Nazarov
Affiliation: Institute of Mechanical Engineering Problems, Bol′shoĭ pr. V. O. 61, 199178 St. Petersburg, Russia
Email: serna@snark.ipme.ru

A. S. Slutskii
Affiliation: Institute of Mechanical Engineering Problems, Bol′shoĭ pr. V. O. 61, 199178 St. Petersburg, Russia
Email: slutskii@snark.ipme.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-06-00937-X
PII: S 1061-0022(06)00937-X
Keywords: Fractal type perforated domain, homogenization, Korn's inequality, corrector
Received by editor(s): September 13, 2004
Published electronically: September 20, 2006
Additional Notes: Supported by RFBR (project no. 03-01-00838)
Article copyright: © Copyright 2006 American Mathematical Society