The problem of Steklov type in a half-cylinder with a small cavity

In the work we consider a Steklov type problem for the Laplace operator in $n$-dimensional cylinder with a small cavity. On the lateral surfaces one of three classic boundary conditions is imposed, the boundary of the cavity is subject to the Dirichlet condition, while on the base of the cylinder we impose the spectral Steklov condition. We prove the convergence theorems for the eigenvalues of this problems as the small parameter, the diameter of the cavity, tends to zero. We construct and justify the complete asymptotic expansions in the small parameter converging both to a simple or a double eigenvalue of the limiting problem, which is the problem without the cavity.