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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.
Hilal Essaouini, Pierre Capodanno. (2020) Analysis of the small oscillations of a heavy almost homogeneous inviscid liquid partially filling an elastic body with negligible density, Punjab University Journal of Mathematics, Volume 52 , Issue No.1.
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