  We consider a boundary-value problem for the second order elliptic
differential operator with rapidly oscillating coefficients in a domain
$\Omega_{\epsilon}$ that is $\epsilon-$periodically perforated by small holes.
The holes are divided into two $\epsilon-$periodical sets depending on the
boundary interaction at their surfaces. Therefore, two different nonlinear
Robin boundary conditions $\sigma_\epsilon (u_\epsilon) + \epsilon \kappa_{m}
(u_\epsilon) = \epsilon g^{(m)}_\epsilon, m=1, 2,$ are given on the
corresponding boundaries of the small holes. The asymptotic analysis of this
problem is made as $\epsilon\to0,$ namely the convergence theorem both for the
solution and for the energy integral is proved without using extension
operators, the asymptotic approximations both for the solution and for the
energy integral are constructed and the corresponding error estimates are
obtained.

Analysis of PDEs