Operator estimates for elliptic equations in multidimensional domains with strongly curved boundaries

A system of semilinear elliptic equations of the second order is considered in a multidimensional domain. The boundary of this domain is curved arbitrarily within a thin layer along the unperturbed boundary. Dirichlet or Neumann conditions are prescribed on the curved boundary. In the case of Neumann conditions certain additional, rather natural and very weak assumptions are made on the structure of the curved boundary. They make it possible to consider a very wide class of curved boundaries, including, for example, classical rapidly oscillating boundaries. It is shown that when the above thin layer shrinks and the curved boundary approaches the unperturbed one, the homogenization of the problem under consideration leads to the same system of equations with the same boundary conditions but imposed on the limit boundary. The main result consists in relevant operator $W_2^1$- and $L_2$-estimates. Bibliography: 29 titles.