About Symmetry in Partially Hinged Composite Plates
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About Symmetry in Partially Hinged Composite Plates Elvise Berchio1
· Alessio Falocchi1
© The Author(s) 2020
Abstract We consider a partially hinged composite plate problem and we investigate qualitative properties, e.g. symmetry and monotonicity, of the eigenfunction corresponding to the density minimizing the first eigenvalue. The analysis is performed by showing related properties of the Green function of the operator and by applying polarization with respect to a fixed plane. As a by-product of the study, we obtain a Hopf type boundary lemma for the operator having its own theoretical interest. The statements are complemented by numerical results. Keywords Composite plate problem · Partially hinged plate · Green function · Polarization Mathematics Subject Classification 35J08 · 35P05 · 74K20
1 Introduction Let = (0, π ) × (−, ) ⊂ R2 with > 0, we consider the weighted eigenvalue problem: ⎧ 2 ⎪ ⎨ u = λ p(x, y)u u(0, y) = u x x (0, y) = u(π, y) = u x x (π, y) = 0 ⎪ ⎩ u yy (x, ±) + σ u x x (x, ±) = u yyy (x, ±) + (2 − σ )u x x y (x, ±) = 0
B
in for y ∈ (−, )
(1.1)
for x ∈ (0, π ) ,
Elvise Berchio [email protected] Alessio Falocchi [email protected]
1
Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129 Torin, Italy
123
Applied Mathematics & Optimization
where σ ∈ [0, 1) and, for α, β ∈ (0, +∞) with α < β fixed, p belongs to the following family of weights: Pα,β :=
∞
p ∈ L () : α ≤ p ≤ β a.e. in and
p d xdy = || .
(1.2)
The interest for problem (1.1) is due to the fact that it describes the oscillating modes of the non-homogeneous partially hinged rectangular plate which, up to scaling, can model the decks of footbridges and suspension bridges, see [6,23,25] for more details; in particular, the partially hinged boundary conditions reflect the fact that decks of bridges are supported by the ground only at the short edges. We also remark that, in this framework, σ represents the so-called Poisson ratio which for most materials belongs to the interval [0, 1), p represents the density function of the plate and the integral condition in (1.2) means that the total mass of the plate is preserved. In order to study the stability properties of the plate it is important to investigate the effect of the density function p on the eigenvalues, i.e on the frequencies of the plate. In this respect, the starting point of the study is the minimization problem: inf
p∈Pα,β
λ1 ( p),
(1.3)
where λ1 ( p) denotes the first eigenvalue of (1.1). There exists a rich literature dealing with the second order Dirichlet version of (1.1)–(1.3) which is usually named composite membrane problem; this corresponds to the problem of building a body of prescribed shape and mass, out of given materials in such a way that the first frequency of the resulting membrane is as small as possible, see e.g. [14–17] and the monograph [27]. In the fourth order case, problem (1.3) is named composite plate problem and has been mainly studied under clamp
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