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Zeitschrift für Analysis und ihre Anwendungen


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Volume 26, Issue 3, 2007, pp. 277–301
DOI: 10.4171/ZAA/1324

Published online: 2007-09-30

Homogenization of Scalar Problem for a Combined Structure with Singular or Thin Reinforcement

Giuseppe Cardone[1], A. Corbo Esposito[2] and S.E. Pastukhova[3]

(1) Università del Sannio, Benevento, Italy
(2) Università degli Studi di Cassino, Italy
(3) Technical University, Moscow, Russian Federation

The homogenization of quadratic integral functionals for combined structures with singular or asymptotically singular reinforcement is studied in a model case in dimension $N=2.$ Generalizations to more general cases in dimension $N=2$ or to some model cases in dimension $N>2$ are discussed. Such results are obtained in the frame of homogenization of problems depending on two parameters developed by V.~V.~Zhikov in %%% here alteration: citations have been expanded. [\textit{Funct. Anal. Appl.} 33 (1999)(1)], [\textit{Sb.~Math.} 191 (2000)(7-8)], and [{\it Izv.~Math.} 66 (2002)(2)]. In particular, an essential tool is the notion of two-scale convergence of sequences of functions belonging to Sobolev spaces with respect to variable measures.

Keywords: Combined structures, singular and thin reinforcement, two-scale convergence, Sobolev spaces with respect to variable measures

Cardone Giuseppe, Corbo Esposito A., Pastukhova S.E.: Homogenization of Scalar Problem for a Combined Structure with Singular or Thin Reinforcement. Z. Anal. Anwend. 26 (2007), 277-301. doi: 10.4171/ZAA/1324