On operator estimates for elliptic equations in two-dimensional domains with fast oscillating boundary and frequent alternation of boundary conditions

A second-order semilinear elliptic equation is considered in an arbitrary two-dimensional domain with boundary that is rapidly oscillating with small amplitude. The oscillations are arbitrary, with no assumption of periodicity or local periodicity. Fast alternating Dirichlet/Neumann boundary conditions are imposed on this boundary. In the case under consideration a Dirichlet problem with the same differential equation arises in the homogenization limit. The main results obtained are $W^1_2$ and $L_2$-operator estimates.