Abstract: The authors study the small oscillations of a heavy viscous liquid in a container closed by an elastic membrane. From the equations of the problem, they obtain its variational formulation, then an operatorial equation in a suitable Hilbert space. By a simple change of the spectral parameter, they get, for the eigenvalues problem, a Krein-Langer pencil, so that they can obtain a few properties of the spectrum. Besides, they prove that the eigenvalues can be obtained by equating to zero a convergent infinite determinant. Finally, they show the existence and the unicity of the solution of the associated evolution problem.
Keywords: Viscous liquid, elastic membrane, variational, operatorial and spectral methods, eigenvalues and evolution problems.
Full text: PDF
Full file: XML