Asymptotic expansion of solutions of a system of elasticity theory in perforated domains

This paper considers the system of elasticity theory with periodic, rapidly oscillating, piecewise continuous coefficients in a domain $\Omega^\varepsilon$, bounded by the hyperplanes $x_n=0$ and $x_n=d$, which contains cavities $G_\varepsilon$ that are periodically distributed (with period $\varepsilon$). For the solutions, periodic in $x_1,\dots,x_{n-1}$, of the system of elasticity theory in the domain $\Omega^\varepsilon\subset\mathbf R^n$ when the displacements are prescribed on the planes $x_n=0$ and $x_n=d$ and the loads on the boundary of $G_\varepsilon$ vanish, an asymptotic expansion in the powers of the parameter $\varepsilon$ is obtained, and the remainder is estimated.
Such problems arise, in particular, in the study of composite materials with a periodic structure, in which every cell consists of finitely many very different materials and includes finitely many cavities, and where the dimension of the cell is characterized by a small parameter $\varepsilon$.
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