Asymptotics of the Spectrum of Variational Problems Arising in the Theory of Fluid Oscillations

This work is a survey of results on the spectral asymptotics of variational problems arising in the theory of small oscillations of a fluid in a vessel near the equilibrium position. These problems were posed by Kopachevsky in the late 1970s and cover various fluid models. The statements of the problems are given both in the form of boundary-value problems in the domain Ω ⊂ \({\mathbb{R}}^{3}\) ${\mathbb{R}}^3$ occupied by the fluid in the equilibrium state and in the form of variational problems on the spectrum of the ratio of quadratic forms. The common features of all the problems under consideration are the presence of an “elliptic” constraint (the Laplace equation for an ideal fluid or a homogeneous Stokes system for a viscous fluid), as well as the occurrence of the spectral parameter in the boundary condition on the free (equilibrium) surface Γ. The spectrum in the considered problems is discrete; the spectral counting functions have power-law asymptotics.