Asymptotic analysis of a boundary-value problem with the nonlinear   boundary multiphase interactions in a perforated domain
  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. 
Taras A. Mel'nyk and Olena A. Sivak