The periodic Euler-Bernoulli equation

Papanicolaou, VG

dc.contributor.author	Papanicolaou, VG	en
dc.date.accessioned	2014-03-01T01:19:37Z
dc.date.available	2014-03-01T01:19:37Z
dc.date.issued	2003	en
dc.identifier.issn	0002-9947	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/15617
dc.subject	Algebraic/geometric multiplicity	en
dc.subject	Beam operator	en
dc.subject	Euler-Bernoulli equation for the vibrating beam	en
dc.subject	Floquet spectrum	en
dc.subject	Hill operator	en
dc.subject	Multipoint eigenvalue problem	en
dc.subject	Pseudospectrum	en
dc.subject.classification	Mathematics	en
dc.subject.other	MATRIX HILLS EQUATION	en
dc.subject.other	SCHRODINGER-OPERATORS	en
dc.subject.other	ANALYTIC PROPERTIES	en
dc.subject.other	ISOSPECTRAL SETS	en
dc.subject.other	SPECTRAL THEORY	en
dc.subject.other	INVERSE PROBLEM	en
dc.subject.other	VIBRATING BEAM	en
dc.subject.other	TRACE FORMULAS	en
dc.subject.other	SYSTEMS	en
dc.subject.other	POTENTIALS	en
dc.title	The periodic Euler-Bernoulli equation	en
heal.type	journalArticle	en
heal.identifier.primary	10.1090/S0002-9947-03-03315-4	en
heal.identifier.secondary	http://dx.doi.org/10.1090/S0002-9947-03-03315-4	en
heal.language	English	en
heal.publicationDate	2003	en
heal.abstract	We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem [a(x) u"(x)]" = lambdarho(x) u(x), -infinity < x < infinity, where the functions a and rho are periodic and strictly positive. This equation models the transverse vibrations of a thin straight ( periodic) beam whose physical characteristics are described by a and rho. Here we develop a theory analogous to the theory of the Hill operator -(d/dx)(2) + q(x). We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or psi-spectrum. Our new analysis begins with a detailed study of the zeros of the function F(lambda; k), for any given "quasimomentum" k is an element of C, where F(lambda; k) = 0 is the Floquet-Bloch variety of the beam equation ( the Hill quantity corresponding to F(lambda; k) is Delta(lambda) -2 cos(kb), where Delta(lambda) is the discriminant and b the period of q). We show that the multiplicity m(lambda) of any zero lambda of F(lambda; k) can be one or two and m(lambda) = 2 (for some k) if and only if lambda is also a zero of another entire function D(lambda), independent of k. Furthermore, we show that D(lambda) has exactly one zero in each gap of the spectrum and two zeros ( counting multiplicities) in each psi-gap. If lambda* is a double zero of F(lambda; k), it may happen that there is only one Floquet solution with quasimomentum k; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree. Next we show that if (alpha; beta) is an open psi-gap of the pseudospectrum (i.e., alpha < β), then the Floquet matrix T(λ) has a specific Jordan anomaly at λ = α and λ = β. We then introduce a multipoint (Dirichlet-type) eigenvalue problem which is the analogue of the Dirichlet problem for the Hill equation. We denote by {μ(n)}(n&ISIN;Z) the eigenvalues of this multipoint problem and show that {μ(n)}(n&ISIN;Z) is also characterized as the set of values of λ for which there is a proper Floquet solution f(x; λ) such that f(0; λ) = 0. We also show ( Theorem 7) that each gap of the L-2(R)-spectrum contains exactly one μ(n) and each ψ-gap of the pseudospectrum contains exactly two μ(n)'s, counting multiplicities. Here when we say "gap" or "ψ-gap" we also include the endpoints (so that when two consecutive bands or - bands touch, the in-between collapsed gap, or ψ-gap, is a point). We believe that {μ(n)}(n&ISIN;Z) can be used to formulate the associated inverse spectral problem. As an application of Theorem 7, we show that if ν* is a collapsed ("closed") ψ-gap, then the Floquet matrix T(ν*) is diagonalizable. Some of the above results were conjectured in our previous works. However, our conjecture that if all the ψ-gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.	en
heal.publisher	AMER MATHEMATICAL SOC	en
heal.journalName	Transactions of the American Mathematical Society	en
dc.identifier.doi	10.1090/S0002-9947-03-03315-4	en
dc.identifier.isi	ISI:000183810800015	en
dc.identifier.volume	355	en
dc.identifier.issue	9	en
dc.identifier.spage	3727	en
dc.identifier.epage	3759	en