Operator Estimates for Problems in Domains with Singularly Curved Boundary: Dirichlet and Neumann Conditions

We consider a system of second-order semilinear elliptic equations in a multidimensional domain with an arbitrarily curved boundary contained in a narrow layer along the unperturbed boundary. The Dirichlet or Neumann condition is imposed on the curved boundary. In the case of the Neumann condition, rather natural and weak conditions are additionally imposed on the structure of the curving. Under these conditions, we show that the homogenized problem is one for the same system of equations in the unperturbed problem with a boundary condition of the same kind as on the perturbed boundary. The main result is operator \(W_{2}^{1}\) - and L₂- estimates.