Convergence of eigenelements of a Steklov-type boundary value problem for the Lame operator in a semi-cylinder with a small cavity

A Steklov-type boundary value problem for the Lame operator in a semi-cylinder containing a small cavity is investigated. The case is considered when an elastic, homogeneous isotropic medium filling a region with a small cavity is rigidly coupled to the lateral boundary of a semi-cylinder and the boundary of a small cavity, which corresponds to a homogeneous Dirichlet boundary condition, and the Steklov spectral condition is set on the basis of the semi-cylinder. The main result consists in proving a theorem on the convergence of the eigenelements of such a singularly perturbed boundary value problem to the eigenelements of the limiting problem (in a semi-cylinder without a cavity) with a small parameter $\varepsilon>$ 0 tending to zero, characterizing the diameter of the cavity. To prove the theorem, a Hilbert space of infinitely differentiable vector functions with a finite Dirichlet integral over a semicylinder was introduced. In contrast to the situation with a limited domain, in the boundary value problem under study, the condition of finiteness of the Dirichlet integral is essential, since it generally ensures finiteness of the norm in the introduced space. The restriction on the finiteness of the Dirichlet integral made it possible to establish a priori estimates that guarantee the uniqueness of solutions to the limiting and perturbed boundary value problems and to establish the equivalence of norms necessary to prove the existence of a solution to the singularly perturbed boundary value problem under study.