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. Frequently alternating Dirichlet and Neumann boundary conditions are imposed on this boundary. In the case under consideration a Dirichlet problem with the same differential equation arises in the limit under the homogenization. The main results obtained are W21-and L2-operator estimates. © 2025 Russian Academy of Sciences, Steklov Mathematical Institute of RAS.