Analysis of the small oscillations of a heavy almost homogeneous inviscid liquid partially filling an elastic body with negligible density

Abstract

In this paper, we study the small oscillations of a system formed by an elastic container with negligible density and a heavy heterogeneous inviscid liquid filling partially the container, in the particular case of an alomost homogeneous liquid, i.e a liquid whose the density in the equilibrium position is practically a linear function of the depth, that differs very little from a constant. By means of an auxiliary problem, that requires a careful study, we reduce the problem to a problem for a liquid only. From the variational formulation of the problem, we obtain its operatorial equations in a suitable Hilbert space. From these, we prove the existence of a spectrum formed by a point spectrum constituted by a countable set of positive real eigenvalues, whose the point of accumulation is the infinity and an essential spectrum filling an interval, that is physically a domain of resonance. Finally, we prove the existence and the unicity of the solution of the associated evolution problem.