Homogenization of the mixed boundary-value problem for a formally selfadjoint elliptic system in a periodically punched domain

Abstract:

A generalized Gårding–Korn inequality is established in a domain $\Omega (h)\subset {\mathbb R}^n$ with a small, of size $O(h)$, periodic perforation, without any restrictions on the shape of the periodicity cell, except for the usual assumptions that the boundary is Lipschitzian, which ensures the Korn inequality in a general domain. Homogenization is performed for a formally selfadjoint elliptic system of second order differential equations with the Dirichlet or Neumann conditions on the outer or inner parts of the boundary, respectively; the data of the problem are assumed to satisfy assumptions of two types: additional smoothness is required from the dependence on either the “slow” variables $x$, or the “fast” variables $y=h^{-1}x$. It is checked that the exponent $\delta \in (0,1/2]$ in the accuracy $O(h^\delta )$ of homogenization depends on the smoothness properties of the problem data.