The Steklov problem in a half-plane: the dependence of eigenvalues on a piecewise-constant coefficient

The Steklov problem considered in the paper describes free two-dimensional oscillations of an ideal, incompressible, heavy fluid in a half-plane covered by a rigid dock with two symmetric gaps. Equivalent reduction of the problem to two spectral problems for integral operators allows us to find limits for all eigenfrequencies as the spacing between gaps tends to both zero and infinity. For the fundamental eigenfrequency and the corresponding eigenfunction two terms are found in the asymptotic expansion as the spacing tends to infinity. It is proved that all eigenvalues are simple for any value of the spacing.