Method of asymptotic partial decomposition of domain and partial homogenization

Application of the method of asymptotic partial decomposition of domain to the following two singularly perturbed boundary value problems is considered. The first one is a boundary value problem for a Poisson equation on a narrow rectangle with the Dirichlet boundary conditions on its smaller sides and the Neumann conditions on the others. The second is a Dirichlet problem in a layer for elliptic operator with coefficients rapidly oscillating with respect to the cross variable.