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 distance between the gaps tends to zero or infinity. For the fundamental eigenfrequency and the corresponding eigenfunction, two terms are found in the asymptotic expansion as the distance tends to infinity. It is proved that all eigenvalues are simple for any distance. Bibliography: 15 titles.Dedicated to the centenary of V. A. Steklovs paper 1]__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 162–190.