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

Abstract

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 [Funct. Anal. Appl. 33 (1999)(1)], [Sb. Math. 191 (2000)(7-8)], and [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.