Operator estimates for weakly non-linear sieve problems with mixed boundary conditions: homogenized Dirichlet condition

Abstract

We consider a boundary value problem for a weakly nonlinear elliptic equation in a multi-dimensional domain, which contains a finely perforated surface. The perforation is arbitrary and non-periodic. We arbitrarily partition each side of surface into two subsets, on one subsect we impose the Dirichlet condition, the other is subject to a weakly nonlinear Robin condition. As a particular case, the mentioned partition includes the one consisting of small pieces, which leads to the frequent alternation of boundary conditions. We formulate quite weak geometric conditions of the perforation of surface and the partitions of its sides, under which the homogenization leads to the Dirichlet condition on both sides of the surface. Our main results are $W_2^1$- and $L_2$-operator estimates.