On estimate of eigenfunctions of the Steklov-type problem with a small parameter in the case of a limit spectrum degeneration

We consider a Steklov-type problem with rapidly alternating boundary conditions (Dirichlet and Steklov) in a bounded two-dimensional domain. The parts of the boundary, where the Dirichlet boundary condition are given, have the length of the order $\varepsilon$ and they alternate with parts of the length of the same order, having the Steklov condition. We prove that the normalized eigenfunctions for a sufficiently small $\varepsilon$ satisfy the Friedrichs-type inequality with the constant of the order $\varepsilon$ and moreover, they converge to zero as $\varepsilon$ tends to zero.