HOMOGENIZATION OF THE MIXED BOUNDARY-VALUE PROBLEM FOR A FORMALLY SELFADJOINT ELLIPTIC SYSTEM IN A PERIODICALLY PUNCHED DOMAIN

A generalized Garding-Korn inequality is established in a domain Omega(h) subset of 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 is an element of (0,1/2] in the accuracy O(h(delta)) of homogenization depends on the smoothness properties of the problem data.