Operator Estimates for the Steklov Problem in an Unbounded Domain with Rapidly Changing Conditions on the Boundary

A spectral problem of the Steklov type for the Laplacian in an unbounded domain with a smooth boundary is considered. The Steklov condition rapidly alternates with the homogeneous Dirichlet condition on a part of the boundary. Operator estimates are obtained, which are used to study the asymptotic behavior of the eigenelements of the original problem as the small parameter tends to zero. The small parameter characterizes the size of the boundary parts with the Dirichlet condition, the distance between which is on the order of the logarithm of the small parameter in a negative power.

Automated summary

The spectral problem of the Steklov type for the Laplacian in an unbounded domain with a smooth boundary is considered, where the Steklov condition rapidly alternates with the homogeneous Dirichlet condition on a part of the boundary. Operator estimates are obtained to study the asymptotic behavior of the eigenelements of the original problem as the small parameter approaches zero. The small parameter characterizes the size of the boundary parts with the Dirichlet condition, with distances between them in order of the logarithm of the small parameter to a negative power.