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A maxmin principle for nonlinear eigenvalue problems with application to a rational spectral problem in fluid-solid vibration. (English) Zbl 1099.35076
Nonlinar eigenvalue problems \(T(\lambda )x =0\) are considered, where for any \(\lambda \in J\), \(T(\lambda )\) is a selfadjoint and bounded operator in a real Hilbert space \(H\), \(J\) is a real open (bounded or unbounded) interval. The author studies a situation when the equation \(\langle T(p(x))x,x\rangle =0\) defines implicitly the Rayleigh functional \(p\) on a subset of \(H\setminus \{0\}\). In the case \(T(\lambda )=\lambda I-A\), the functional \(p\) coincides with the usual Rayleigh quotient. A maxmin principle for the Rayleigh functional \(p\) characterizing the \(n\)-th eigenvalue is given. The result is a generalization of the maxmin characterization of R. Courant, E. Fischer and H. Weyl for linear eigenvalue problems. It is applied to a location of eigenvalues of a rational eigenvalue problem governing free vibrations of a tube bundle immersed in a slightly compressible fluid under certain simplifying assumptions.

MSC:
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
76M30 Variational methods applied to problems in fluid mechanics
49S05 Variational principles of physics
49R50 Variational methods for eigenvalues of operators (MSC2000)
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