The periodic Euler-Bernoulli equation
Author: Vassilis G. Papanicolaou
Journal: Trans. Amer. Math. Soc. 355 (2003), 3727-3759
MSC (2000): Primary 34B05, 34B10, 34B30, 34L40, 74B05
Published electronically: May 29, 2003
MathSciNet review: 1990171
Full-text PDF Free Access
Abstract: We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem
where the functions and are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described by and . Here we develop a theory analogous to the theory of the Hill operator .
We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or -spectrum.
Our new analysis begins with a detailed study of the zeros of the function , for any given ``quasimomentum'' , where is the Floquet-Bloch variety of the beam equation (the Hill quantity corresponding to is , where is the discriminant and the period of ). We show that the multiplicity of any zero of can be one or two and (for some ) if and only if is also a zero of another entire function , independent of . Furthermore, we show that has exactly one zero in each gap of the spectrum and two zeros (counting multiplicities) in each -gap. If is a double zero of , it may happen that there is only one Floquet solution with quasimomentum ; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree.
Next we show that if is an open -gap of the pseudospectrum (i.e., ), then the Floquet matrix has a specific Jordan anomaly at and .
We then introduce a multipoint (Dirichlet-type) eigenvalue problem which is the analogue of the Dirichlet problem for the Hill equation. We denote by the eigenvalues of this multipoint problem and show that is also characterized as the set of values of for which there is a proper Floquet solution such that .
We also show (Theorem 7) that each gap of the -spectrum contains exactly one and each -gap of the pseudospectrum contains exactly two 's, counting multiplicities. Here when we say ``gap'' or ``-gap'' we also include the endpoints (so that when two consecutive bands or -bands touch, the in-between collapsed gap, or -gap, is a point). We believe that can be used to formulate the associated inverse spectral problem.
As an application of Theorem 7, we show that if is a collapsed (``closed'') -gap, then the Floquet matrix is diagonalizable.
Some of the above results were conjectured in our previous works. However, our conjecture that if all the -gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.
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Vassilis G. Papanicolaou
Affiliation: Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67260-0033
Address at time of publication: Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80, Athens, Greece
Keywords: Euler-Bernoulli equation for the vibrating beam, beam operator, Hill operator, Floquet spectrum, pseudospectrum, algebraic/geometric multiplicity, multipoint eigenvalue problem
Received by editor(s): November 13, 2001
Received by editor(s) in revised form: November 10, 2002
Published electronically: May 29, 2003
Article copyright: © Copyright 2003 American Mathematical Society