Homogenization of an elliptic system under condensing perforation of the domain

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 $\varkappa\in(0,1/2]$.