# The periodic Euler-Bernoulli equation

@article{Papanicolaou2003ThePE,
title={The periodic Euler-Bernoulli equation},
author={Vassilis G. Papanicolaou},
journal={Transactions of the American Mathematical Society},
year={2003},
volume={355},
pages={3727-3759}
}
• V. Papanicolaou
• Published 29 May 2003
• Mathematics
• Transactions of the American Mathematical Society
We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem [a(x)u(x)] = λρ(x)u(x), -∞ < x < ∞, where the functions a and p 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 p. 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…
An inverse spectral result for the periodic Euler-Bernoulli equation
The Floquet (direct spectral) theory of the periodic Euler-Bernoulli equation has been developed by the author in [19], [21], and [20]. Here we begin a systematic study of the inverse periodic
The inverse periodic spectral theory of the Euler-Bernoulli equation
The Floquet (direct spectral) theory of the periodic Euler-Bernoulli equation has been developed by the author in [37], [41], and [38]. A particular case of the inverse problem has been studied in
Spectral estimates for a periodic fourth-order operator
• Mathematics
• 2008
We consider the operator $H={d^4dt^4}+{ddt}p{ddt}+q$ with 1-periodic coefficients on the real line. The spectrum of $H$ is absolutely continuous and consists of intervals separated by gaps. We
Spectral asymptotics for periodic fourth-order operators
• Mathematics
• 2005
We consider the operator d 4 dt4 +V on the real line with a real periodic potential V . The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define a
Spectral analysis for periodic solutions of the Cahn–Hilliard equation on R
. We consider the spectrum associated with the linear operator obtained when the Cahn–Hilliard equation on R is linearized about a stationary periodic solution. Our analysis is particularly motivated
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We consider the spectrum associated with the linear operator obtained when the Cahn–Hilliard equation on $${\mathbb{R}}$$ is linearized about a stationary periodic solution. Our analysis is
Even order periodic operators on the real line
• Mathematics
• 2010
We consider $2p\ge 4$ order differential operator on the real line with a periodic coefficients. The spectrum of this operator is absolutely continuous and is a union of spectral bands separated by
Spectral analysis of stationary solutions of the Cahn--Hilliard equation
For the Cahn–Hilliard equation on R, there are precisely three types of bounded non-constant stationary solutions: periodic solutions, pulse-type reversal solutions, and monotonic transition waves.
A Survey on Extremal Problems of Eigenvalues
• Mathematics
• 2012
Given an integrable potential , the Dirichlet and the Neumann eigenvalues and of the Sturm-Liouville operator with the potential q are defined in an implicit way. In recent years, the authors and

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