On Operator Estimates for Elliptic Operators with Mixed Boundary Conditions in Two-Dimensional Domains with Fast Oscillating Boundary

We consider a second-order elliptic operator with sufficiently smooth variable coefficients in an arbitrary two-dimensional domain with fast oscillating boundary under the assumption that the oscillation amplitude is small. The structure of the oscillations is fairly arbitrary and 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 succeed to modify the technique from our previous papers appropriately and to prove the uniform resolvent convergence of the perturbed operator to the homogenized operator and estimate the convergence rate.