On operator estimates for elliptic operators with mixed boundary conditions in two-dimensional domains with rapidly oscillating boundary

The paper considers a second-order elliptic operator with variable sufficiently smooth coefficients in an arbitrary two-dimensional domain with rapidly oscillating boundary under the assumption that the oscillation amplitude is small. The structure of the oscillations is fairly arbitrary in that no periodicity or local periodicity conditions are imposed. The oscillating boundary is divided into two components with the Dirichlet boundary condition posed on one of the components and the Neumann condition, on the other. Such mixed boundary conditions are preserved under homogenization; as a result, the functions in the domain of the homogenized operator have weak power-law singularities. Despite these singularities, we have been able to modify the technique in our previous papers appropriately so as to prove the uniform resolvent convergence of the perturbed operator to the homogenized operator and estimate the convergence rate.