On resolvent of multi-dimensional operators with frequent alternation of boundary conditions: critical case

We consider an elliptic operator in a multi-dimensional domain with frequent alternation of Dirichlet and Robin conditions. We study the case, when the homogenized operator has Robin condition with an additional coefficient generated by the geometry of the alternation. We prove the norm resolvent convergence of the perturbed operator to the homogenized one and obtain the estimate for the rate of convergence. We construct the complete asymptotic expansion for the resolvent in the case, when it acts on sufficiently smooth functions.