Operator estimates for problems in domains with singularly curved boundary: Dirichlet and Neumann conditions

We consider a system of second order semi-linear elliptic equations in a multidimensional domain, the boundary of which is arbitrarily curved and is contained in a narrow layer along the unperturbed boundary. On the curve boundary we impose the Dirichlet or Neumann condition. In the case of the Neumann condition, on the structure of curving we additionally impose rather natural and weak conditions. Under such conditions we show that the homogenized problem is for the same system of equations in the unperturbed problem with the boundary condition of the same kind. The main result are $W_2^1$- and $L_2$ -operator estimates.