Asymptotics and Estimates of the Convergence Rate in a Three-Dimensional Boundary-Value Problem with Rapidly Alternating Boundary Conditions

摘要

We consider a singularly perturbed boundary-value eigenvalue problem for the Laplace operator in a cylinder with rapidly alternating type of the boundary condition on the lateral surface. The change of the boundary conditions is realized by splitting the lateral surface into many narrow strips on which the Dirichlet and Neumann conditions alternate. We study the case in which the averaged problem contains the Dirichlet boundary condition on the lateral surface. In the case of strips with slowly varying width we construct the first terms of the asymptotic expansions of eigenfunctions; moreover, in the case of strips with rapidly varying width we obtain estimates for the convergence rate.