On the first frequency of reinforced partially hinged plates
Elvise Berchio, Alessio Falocchi, Alberto Ferrero, Debdip Ganguly
OON THE FIRST FREQUENCY OF REINFORCEDPARTIALLY HINGED PLATES
ELVISE BERCHIO, ALESSIO FALOCCHI, ALBERTO FERRERO, AND DEBDIP GANGULY
Abstract.
We consider a partially hinged rectangular plate and its normal modes. The dynamicalproperties of the plate are influenced by the spectrum of the associated eigenvalue problem. In order toimprove the stability of the plate, it seems reasonable to place a certain amount of stiffening materialin appropriate regions. If we look at the partial differential equation appearing in the model, thiscorresponds to insert a suitable weight coefficient inside the equation. A possible way to locate suchregions is to study the eigenvalue problem associated to the aforementioned weighted equation. In thepresent paper we focus our attention essentially on the first eigenvalue and on its minimization in termsof the weight. We prove the existence of minimizing weights inside special classes and we try to describethem together with the corresponding eigenfunctions. Introduction
Following [15] one may view a bridge as a long narrow rectangular thin plate Ω hinged at two oppositeedges and free on the remaining two edges: this plate well describes decks of footbridges and suspensionbridges which, at the short edges, are supported by the ground. We refer to the monograph [16] for adetailed survey either of old and new mathematical models for suspension bridges. Up to scaling, wemay assume that the plate has length π and width 2 (cid:96) with 2 (cid:96) (cid:28) π so thatΩ = (0 , π ) × ( − (cid:96), (cid:96) ) ⊂ R . There is a growing interest of engineers on the shape optimization for the design of bridges and, inparticular, on the sensitivity analysis of certain eigenvalue problems, see [18, Chapter 6]. As pointed outby Banerjee [3], the free vibration analysis is a fundamental pre-requisite before carrying out a flutteranalysis . Whence, in the the stability analysis of the plate a central role is played by the followingeigenvalue problem:(1) ∆ u = λ u in Ω u (0 , y ) = u xx (0 , y ) = u ( π, y ) = u xx ( π, y ) = 0 for y ∈ ( − (cid:96), (cid:96) ) u yy ( x, ± (cid:96) ) + σu xx ( x, ± (cid:96) ) = u yyy ( x, ± (cid:96) ) + (2 − σ ) u xxy ( x, ± (cid:96) ) = 0 for x ∈ (0 , π ) , where σ denotes the Poisson ratio of the material forming the plate. For most elastic materials onehas 0 < σ < .
5; since we aim to model the deck of a bridge, which is a mixture of concrete and steel,one may take σ = 0 .
2. The boundary conditions on the short edges tell that the plate is hinged; theseconditions are named Navier since their first appearance in [22]. We refer to [4] for the derivation of (1)from the total energy of the plate. Note that in [15] the whole spectrum of (2) was determined, whilein [5] the results were exploited to study the so-called torsional stability of suspension bridges for smallenergies. Furthermore, in [4] the variation of the eigenvalues, under domain deformations, which maynot preserve the area, was investigated, see also [20] for related results about Dirichlet polyharmoniceigenvalue problems.In order to improve the stability of the plate, one may think to place a certain amount of stiff materialwithin the plate. In mathematical terms this can be modelled by inserting into the equation a weight p , Date : April 22, 2019.2010
Mathematics Subject Classification.
Key words and phrases. eigenvalues; plates; torsional instability; suspension bridges. a r X i v : . [ m a t h . A P ] A p r ELVISE BERCHIO, ALESSIO FALOCCHI, ALBERTO FERRERO, AND DEBDIP GANGULY properly chosen to describe the action of the reinforcement and we end up with the weighted eigenvalueproblem:(2) ∆ u = λ p ( x, y ) u in Ω u (0 , y ) = u xx (0 , y ) = u ( π, y ) = u xx ( π, y ) = 0 for y ∈ ( − (cid:96), (cid:96) ) u yy ( x, ± (cid:96) ) + σu xx ( x, ± (cid:96) ) = u yyy ( x, ± (cid:96) ) + (2 − σ ) u xxy ( x, ± (cid:96) ) = 0 for x ∈ (0 , π ) , where, for 0 < α (cid:54) β fixed, p belongs to the following family of weights(3) P α,β := (cid:26) p ∈ L ∞ (Ω) : α (cid:54) p (cid:54) β a.e. in Ω and (cid:90) Ω p dxdy = | Ω | (cid:27) . The spectral analysis of (2) should indicate where to place the stiff material within the plate. Inthis respect, the condition on the integral of p is posed in order to make the comparison with the case p ≡ u = f ( x, y )1 + dχ D ( x, y ) in Ωsubject to the boundary conditions in (2), where χ D is the characteristic function of D ⊂ Ω and d > u of this equation describesthe vertical displacement of the plate under the action of a load f while the weight p is here explicitlygiven by p ( x, y ) = 1 / (1 + dχ D ( x, y )). In particular, p can be seen as an “aerodynamic damper” placedin D in order to reduce the action of the external force f . Hence, the lowest are the values of p in someregion of the rectangle Ω, the highest is the amount of stiffening material placed in that region, andthe lower bound p (cid:62) α > p onthe fundamental frequency λ ( p ), namely to study:inf p ∈ P α,β λ ( p ) . When λ ( p ) is the first weighted eigenvalue of − ∆ under Dirichlet boundary conditions, the aboveproblem coincides with the so-called composite membrane problem, see [7]-[10],[24], while if λ ( p ) is thefirst weighted eigenvalue of ∆ under Dirichlet or Navier boundary conditions, it becomes the compositeplate problem, see [1],[2],[11]-[14]. In this field of research, typical results are existence of optimal pairsand their qualitative properties, such as symmetry or symmetry breaking. From this point of view acrucial obstruction, when passing from the membrane to the plate problem, namely from the second tothe fourth order case, is represented by the loss of maximum and comparison principles which usuallyenter either in the study of the simplicity of the first eigenvalue and in the techniques applied to provesymmetry results, such as reflections methods or moving planes techniques. Nevertheless, a suitablechoice of the boundary conditions (e.g. Navier or Steklov b.c.) or of the geometry of the domain(e.g. small perturbations of balls) may yield the validity of so-called positivity preserving property which basically means that solutions, of the associated linear problem, maintain the sign of data. Thisproperty generally allows to extend some of the results known in the second order to the higher ordercase. As concerns problem (2), the difficulties when passing to the higher order, are even increased bythe choice of the unusual boundary conditions for which no positivity preserving property is known.Note that, problem (2) with p (cid:54)≡ y variable, that we prove in Theorem 3.7 below, having its own theoretical interest.We note that the above mentioned restriction on admissible weights is also justified by the applicativeorigin of our problem. Indeed, it is known that minimization problems, like the composite membrane ONHOMOGENEOUS PLATES 3 problem, naturally lead to homogenization [21], see also [19] for a stiffening problem for the torsion ofa bar. Homogenization would lead to optimal designs with reinforcements scattered throughout thestructure, namely designs impossible to implement for engineers. Hence, to avoid homogenization, theclass of admissible reinforcements should be sufficiently small. See also Nazarov-Sweers-Slutskij [23],where only “macro” reinforcements are considered, although in a fairly different setting.The paper is organised as follows. Section 2 is devoted to the description of the notations and ofsome results about the case p ≡
1. In Section 3 one can find the main results of the paper whichare proved in Sections 4 and 5. In Section 6 we show some numerical results on the behaviour of theeigenvalues which complement our theoretical analysis. Finally, in Section 7 we show the validity ofa positivity preserving property for a one dimensional fourth order problem, coming from a suitableFourier decomposition of solutions to the plate problem.2.
Notations and known results when p ≡ H ∗ (Ω) = (cid:8) u ∈ H (Ω) : u = 0 on { , π } × ( − (cid:96), (cid:96) ) (cid:9) . For any σ ∈ (0 , H ∗ (Ω) is a Hilbert space when endowed with the scalar product( u, v ) H ∗ := (cid:90) Ω [∆ u ∆ v + (1 − σ )(2 u xy v xy − u xx v yy − u yy v xx )] dx dy and associated norm (cid:107) u (cid:107) H ∗ (Ω) = ( u, u ) H ∗ (Ω) , which is equivalent to the usual norm in H (Ω), see [15, Lemma 4.1]. From now onward we assume σ ∈ (0 ,
1) fixed. Then problem (2) may also be formulated in the following weak sense(4) ( u, v ) H ∗ (Ω) = λ (cid:90) Ω p ( x, y ) uv dx dy ∀ v ∈ H ∗ (Ω) , where, for 0 < α (cid:54) β fixed, p belongs to the family of weights P α,β defined in (3). Clearly, the constantweight p ≡ P , . Since the bilinear form ( u, v ) H ∗ is continuous and coercive and p ∈ L ∞ (Ω) is positive a.e. in Ω, standard spectral theory of self-adjoint operators then shows that theeigenvalues of (2) may be ordered in an increasing sequence of strictly positive numbers diverging to+ ∞ and that the corresponding eigenfunctions form a complete system in H ∗ (Ω).Since p ∈ L ∞ (Ω) , by elliptic regularity the eigenfunctions are at least in C (Ω) . Furthermore, thefirst eigenvalue is characterized by(5) λ ( p ) := inf u ∈ H ∗ (Ω) \{ } (cid:107) u (cid:107) H ∗ (cid:107)√ p u (cid:107) . When p ≡ Proposition 2.1.
Let p ≡ in (2) . The set of eigenvalues of (2) may be ordered in an increasingsequence of strictly positive numbers diverging to + ∞ and any eigenfunction belongs to C ∞ (Ω) ; the setof eigenfunctions of (2) is a complete system in H ∗ (Ω) . Moreover: ( i ) for any m (cid:62) , there exists a unique eigenvalue λ = µ m, ∈ ((1 − σ ) m , m ) with correspondingeigenfunction (cid:2) µ / m, − (1 − σ ) m (cid:3) cosh (cid:16) y (cid:113) m + µ / m, (cid:17) cosh (cid:16) (cid:96) (cid:113) m + µ / m, (cid:17) + (cid:2) µ / m, + (1 − σ ) m (cid:3) cosh (cid:16) y (cid:113) m − µ / m, (cid:17) cosh (cid:16) (cid:96) (cid:113) m − µ / m, (cid:17) sin( mx ) ;( ii ) for any m (cid:62) and any k (cid:62) there exists a unique eigenvalue λ = µ m,k > m satisfying (cid:16) m + π (cid:96) (cid:0) k − (cid:1) (cid:17) < µ m,k < (cid:16) m + π (cid:96) ( k − (cid:17) and with corresponding eigenfunction (cid:2) µ / m,k − (1 − σ ) m (cid:3) cosh (cid:16) y (cid:113) µ / m,k + m (cid:17) cosh (cid:16) (cid:96) (cid:113) µ / m,k + m (cid:17) + (cid:2) µ / m,k + (1 − σ ) m (cid:3) cos (cid:16) y (cid:113) µ / m,k − m (cid:17) cos (cid:16) (cid:96) (cid:113) µ / m,k − m (cid:17) sin( mx ) ;( iii ) for any m (cid:62) and any k (cid:62) there exists a unique eigenvalue λ = ν m,k > m with correspondingeigenfunctions (cid:2) ν / m,k − (1 − σ ) m (cid:3) sinh (cid:16) y (cid:113) ν / m,k + m (cid:17) sinh (cid:16) (cid:96) (cid:113) ν / m,k + m (cid:17) + (cid:2) ν / m,k + (1 − σ ) m (cid:3) sin (cid:16) y (cid:113) ν / m,k − m (cid:17) sin (cid:16) (cid:96) (cid:113) ν / m,k − m (cid:17) sin( mx ) ;( iv ) for any m (cid:62) satisfying (cid:96)m √ (cid:96)m √ > (cid:0) − σσ (cid:1) there exists a unique eigenvalue λ = ν m, ∈ ( µ m, , m ) with corresponding eigenfunction (cid:2) ν / m, − (1 − σ ) m (cid:3) sinh (cid:16) y (cid:113) m + ν / m, (cid:17) sinh (cid:16) (cid:96) (cid:113) m + ν / m, (cid:17) + (cid:2) ν / m, + (1 − σ ) m (cid:3) sinh (cid:16) y (cid:113) m − ν / m, (cid:17) sinh (cid:16) (cid:96) (cid:113) m − ν / m, (cid:17) sin( mx ) . Finally, if (6) the unique positive solution s > of: tanh( √ s(cid:96) ) = (cid:18) σ − σ (cid:19) √ s(cid:96) is not an integer,then the only eigenvalues are the ones given in ( i ) − ( iv ) . In the following, to avoid too many distinctions, we will always assume that (6) holds.By Proposition 2.1 and [15, Section 7] it is readily deduced that the first eigenvalue of problem (2)with p ≡ µ , , namely λ (1) = µ , , it is simple and up to constant multiplier the first eigenfunctionis given by(7) u ( x, y ) = (cid:2) µ / , − (1 − σ ) (cid:3) cosh (cid:16) y (cid:113) µ / , (cid:17) cosh (cid:16) (cid:96) (cid:113) µ / , (cid:17) + (cid:2) µ / , + (1 − σ ) (cid:3) cosh (cid:16) y (cid:113) − µ / , (cid:17) cosh (cid:16) (cid:96) (cid:113) − µ / , (cid:17) sin x . Hence, u is positive in Ω, convex in the y − variable and concave in the x − variable.3. Main results
Let 0 < α < β be two fixed constants and let P α,β be the class of admissible weights defined inSection 1. Then, clearly α (cid:54) β (cid:62) . Recalling (5), we focus on the double infimum problem(8) λ α,β := inf p ∈ P α,β λ ( p ) = inf p ∈ P α,β inf u ∈ H ∗ (Ω) \{ } (cid:107) u (cid:107) H ∗ (Ω) (cid:107)√ p u (cid:107) . Definition 3.1.
A couple ( u p , p ) ∈ H ∗ (Ω) × P α,β which realises the double infimum in (8) is called an optimal pair . Adapting to our case [9, Theorem 13] and [11, Theorem 1.4], it can be shown that there exists anoptimal pair ( u p , p ) for problem (8) and u p and p are suitably related. Theorem 3.2.
For every < α < β , there exists and optimal pair ( u p , p ) ∈ H ∗ (Ω) × P α,β . Furthermore, u p and p are related as follows (9) p ( x, y ) = αχ S ( x, y ) + βχ Ω \ S ( x, y ) for a.e. ( x, y ) ∈ Ω , where χ S and χ Ω \ S are the characteristic functions of the sets S and Ω \ S and S ⊂ Ω is such that | S | = β − β − α | Ω | and S = { ( x, y ) ∈ Ω : u p ( x, y ) (cid:54) t } for some t (cid:62) . ONHOMOGENEOUS PLATES 5
Note that since Ω is planar, u p ∈ C (Ω) and the set S is closed. The above result suggests that theplate can be made out of two materials but it gives no informations about the location of the materialsand hence, no practical informations on how to built the plate. To this aim, a more explicit suggestion,even if more rought, is provided by the following Proposition 3.3.
Let < α < β and p ∈ P α,β satisfy one of the following assumptions (i) p = p ( y ) is even and there exists z ∈ (0 , (cid:96) ) such that p ( y ) (cid:54) for y ∈ [0 , z ] and p ( y ) (cid:62) for y ∈ [ z, (cid:96) ) . (ii) p = p ( x ) is symmetric with respect to the line x = π and there exists s ∈ (0 , π ) such that p ( x ) (cid:54) for x ∈ (0 , s ] and p ( x ) (cid:62) for x ∈ [ s, π . Then, (10) λ ( p ) (cid:54) λ (1) = µ , , where the µ , is as defined in Proposition 2.1-(i). Remark 3.4.
It’s worth noting that the same idea of the proof of Proposition 3.3-(i) can be repeatedto prove that (10) holds if p ∈ P α,β satisfies (iii) p = p ( y ) is even and there exist N + 2 points y < y < y < ... < y N +2 = (cid:96) such that p ( y ) (cid:54) for y ∈ [ y h , y h +1 ] , p ( y ) (cid:62) for y ∈ [ y h +1 , y h +2 ] and (cid:90) y h +2 y h ( p − dy = 0 , for all h = 0 , ..., N . Since the weights considered in Proposition 3.3 prove to be effective in decreasing the first frequencyof (1), by combining Proposition 3.3 with Theorem 3.2, it is reasonable to include in the list of candidatesolutions to problem (8) the weights:(11) p ( y ) = αχ ( − (cid:96) ( β − β − α , (cid:96) ( β − β − α ) ( y ) + βχ ( − (cid:96),(cid:96) ) \ ( − (cid:96) ( β − β − α , (cid:96) ( β − β − α ) ( y ) y ∈ ( − (cid:96), (cid:96) )and p ( x ) = βχ ( π − αβ − α , π β − − αβ − α ) ( x ) + αχ (0 ,π ) \ ( π − αβ − α , π β − − αβ − α ) ( x ) x ∈ (0 , π ) . Figure 1.
On the left, plot of the eigenfunction u ,p ( x, y ), corresponding to λ ( p ) with p ( y ) as in (11), intersected with t >
0. On the right, plot of p ( y ) (top) and plot of thesublevel set S = { ( x, y ) ∈ Ω : u ,p ( x, y ) (cid:54) t } (bottom). ELVISE BERCHIO, ALESSIO FALOCCHI, ALBERTO FERRERO, AND DEBDIP GANGULY
In Section 6 we obtained numerically a positive eigenfunction, denoted by u ,p ( x, y ), correspondingto λ ( p ) with p ( y ) as in (11). In Figure 1 on the left, we plot z = u ,p ( x, y ) and we use it to determinequalitatively what should be the set S predicted by Theorem 3.2. A comparison between the weight p ( x, y ) in (9), with this choice of the set S , and the weight p ( y ) in (11) is shown in Figure 1 on theright. From these plots we infer that ( u ,p ( x, y ) , p ( y )) cannot belong to a theoretical optimal pair of(8).On the other hand, when restricting the class of admissible weights to a suitable subset of P α,β , inTheorem 3.5 below we prove that indeed p ( y ) belongs to an optimal pair provided that the constant β satisfies a suitable upper bound. Note that the numerical results we state in Section 6 suggests thatthis upper bound is merely a technical condition. Theorem 3.5.
Let < α < β < min { /µ , , (1 − σ )2 } and denote P α,β = { p ∈ P α,β : p = p ( y ) is even, p is piecewise continuous in ( − (cid:96), (cid:96) ) and ∃ z ∈ (0 , (cid:96) ) : p ( y ) (cid:54) in [0 , z ] , p ( y ) (cid:62) in [ z, (cid:96) ) } . The following statements hold: ( i ) if p , p ∈ P α,β and there exists z ∈ (0 , (cid:96) ) such that p ( y ) (cid:54) p ( y ) in [0 , z ] and p ( y ) (cid:62) p ( y ) in [ z, (cid:96) ) , then λ ( p ) (cid:54) λ ( p ) ;( ii ) we have min p ∈ P α,β λ ( p ) = λ ( p ) , where p is as defined in (11) . It is worth noting that, in order to lower the first eigenvalue of ∆ under Dirichlet or Navier boundaryconditions, since the eigenfunctions vanish on the boundary, one expects that the weight is more effectiveif it achieves its lowest value close to the boundary, see e.g. [11, Theorem 1.5]. Theorem 3.5 shows thatthe partially hinged boundary conditions lead to a complete different situation since the weight p ( y )achieves its lowest value α far from the free long edges, see Figure 1 on the right (top). This behaviouris somehow related to the monotonicity of the first eigenfunction, as shown by Theorem 3.6 below, cfr.Figure 2. Theorem 3.6.
Let < α < β < min { /µ , , (1 − σ )2 } and let P α,β be the family of weights definedin Theorem 3.5. Then, for any p ∈ P α,β the first eigenvalue λ ( p ) of (4) is simple. Furthermore, if u ,p is an eigenfunction of λ ( p ) then u ,p is of one sign in Ω and moreover u ,p can be written as u ,p ( x, y ) = ϕ ,p ( y ) sin( x ) with ϕ ,p ( y ) even and strictly monotone in (0 , (cid:96) ) . Unfortunately, the above statement does not carry over to all weights p ∈ P α,β . This is related tothe well-know loss of comparison principles for higher order elliptic operators. Indeed, the proof ofTheorem 3.6 highly relies on a sort of restricted positivity preserving property with respect to the y variable that we prove by separating variables. More precisely, we have Theorem 3.7.
Let m (cid:62) be an integer and σ ∈ (0 , . Furthermore, let u ∈ H ∗ (Ω) be a weak solutionto the problem ∆ u = f ( y ) sin( mx ) in Ω u (0 , y ) = u xx (0 , y ) = u ( π, y ) = u xx ( π, y ) = 0 for y ∈ ( − (cid:96), (cid:96) ) u yy ( x, ± (cid:96) ) + σu xx ( x, ± (cid:96) ) = u yyy ( x, ± (cid:96) ) + (2 − σ ) u xxy ( x, ± (cid:96) ) = 0 for x ∈ (0 , π ) , ONHOMOGENEOUS PLATES 7
Figure 2.
Qualitative plot of u ,p ( x, y ) = ϕ ,p ( y ) sin( x ). namely ( u, v ) H ∗ = (cid:90) Ω f ( y ) sin( mx ) v ∀ v ∈ H ∗ (Ω) . Then, u ( x, y ) = w m ( y ) sin( mx ) and the following implication holds f (cid:62) in ( − (cid:96), (cid:96) ) ( f (cid:54)≡ ⇒ w m ( y ) > in [ − (cid:96), (cid:96) ] . Proof of Theorem 3.2 and Proposition 3.3
Proof of Theorem 3.2.
We start with the existence issue.
Lemma 4.1.
For every < α < β , the double infimum in (8) is achieved.Proof. Let { p m } m ⊂ P α,β be a minimizing sequence for λ α,β , i.e. λ ( p m ) = λ α,β + o (1) as m → ∞ . Let now u p m ∈ H ∗ (Ω) be a (normalized) eigenfunction to λ ( p m ), namely λ ( p m ) = || u p m || H ∗ (Ω) and (cid:82) Ω p m u p m dx dy = 1 . This immediately implies || u p m || H ∗ (cid:54) C, for some positive constant C. Therefore,using the compact embedding of H ∗ (Ω) (cid:44) → L (Ω) , we can extract two subsequences, still denoted by u p m , such that u p m (cid:42) u in H ∗ (Ω) as m → ∞ ,u p m → u in L (Ω) as m → ∞ . Moreover, p m ∈ P α,β implies || p m || L ∞ (Ω) (cid:54) β and therefore up to a subsequence, p m (cid:42) p in L (Ω) as m → ∞ . By this we have that | Ω | = (cid:82) Ω p m dx dy = (cid:82) Ω p dx dy + o (1) as m → ∞ and, since stronglyclosed convex sets are weakly closed, that α (cid:54) p (cid:54) β a.e. in Ω. Hence, p ∈ P α,β . On the other hand,we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω ( p m u p m − p u ) dx dy (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Ω p m ( u p m − u ) dx dy + (cid:90) Ω u ( p m − p ) dx dy (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) β (cid:90) Ω | ( u p m − u )( u p m + u ) | dx dy + (cid:107) u (cid:107) L ∞ (Ω) (cid:90) Ω | u | | p m − p | dx dy = 2 β (cid:107) u (cid:107) L (Ω) (cid:107) u p m − u (cid:107) L (Ω) + o (1) = o (1) as m → ∞ , where we have exploited the fact that H ∗ (Ω) ⊂ L ∞ (Ω) since Ω is a planar domain. Hence, we concludethat (cid:82) Ω p u dx dy = 1 . Furthermore,
ELVISE BERCHIO, ALESSIO FALOCCHI, ALBERTO FERRERO, AND DEBDIP GANGULY λ ( p ) (cid:54) || u || H ∗ (cid:54) lim inf m →∞ || u p m (cid:107) H ∗ = λ α,β . Hence λ α,β (cid:54) λ ( p ) = || u || H ∗ (cid:54) λ α,β . Therefore, the couple ( p, u ) is an optimal pair. Hence, u p = u and this completes the proof. (cid:3) To problem (8) we associate the following double infimum problem(12) Λ α,β := inf η ∈ N α,β inf u ∈ H ∗ (Ω) \{ } || u || H ∗ (Ω) + λ α,β ( β − α ) (cid:82) Ω η u dx dy (cid:82) Ω u dx dy , where λ α,β is as in (8) and N α,β = (cid:26) η ∈ L ∞ (Ω) : 0 (cid:54) η (cid:54) (cid:90) Ω η dx dy = β − β − α | Ω | (cid:27) . The proof of Lemma 4.1 with minor changes shows that also problem (12) admits an optimal pair( u η , η ) ∈ H ∗ (Ω) × N α,β . Furthermore, there is an one-to-one correspondence between problems (8) and(12). Indeed, to any η ∈ N α,β we can associate p η ∈ P α,β by setting p η = β − η ( β − α ) . Clearly α (cid:54) p η (cid:54) β and (cid:90) Ω p η dx dy = β | Ω | − ( β − α ) (cid:90) Ω η dx dy = | Ω | . Viceversa to any p ∈ P α,β we can associate η p ∈ N α,β by setting η p = β − pβ − α . Clearly 0 (cid:54) η (cid:54) (cid:82) Ω η p dx dy = β − β − α | Ω | . Furthermore, we have
Lemma 4.2.
Let λ α,β and Λ α,β be as defined in (8) and in (12) . There holds Λ α,β = λ α,β β. Proof.
We shall prove the lemma in two steps.
Step 1 :
Let p ∈ P α,β and u p ∈ H ∗ (Ω) such that λ α,β is achieved for this optimal pair and let η p = β − pβ − α ∈ N α,β . Clearly we haveΛ α,β (cid:54) inf u ∈ H ∗ (Ω) \{ } || u || H ∗ (Ω) + λ α,β ( β − α ) (cid:82) Ω η p u dx dy (cid:82) Ω u dx dy = inf u ∈ H ∗ (Ω) \{ } || u || H ∗ (Ω) − λ α,β (cid:82) Ω p u dx dy + λ α,β β (cid:82) Ω u dx dy (cid:82) Ω u dx dy (cid:54) || u p || H ∗ (Ω) − λ α,β (cid:82) Ω p u p dx dy (cid:82) Ω u p dx dy (cid:124) (cid:123)(cid:122) (cid:125) =0 + λ α,β β = λ α,β β. ONHOMOGENEOUS PLATES 9
Step 2 :
Let now η ∈ N α,β and p η ∈ P α,β with η = β − p η β − α , i.e., p η = β − η ( β − α ) . Then for any u ∈ H ∗ (Ω) \ { }|| u || H ∗ (Ω) + λ α,β ( β − α ) (cid:82) Ω η u dx dy (cid:82) Ω u dx dy = || u || H ∗ (Ω) − λ α,β (cid:82) Ω p η u dx dy + λ α,β β (cid:82) Ω u dx dy (cid:82) Ω u dx dy . (13)Since, p η ∈ P α,β implies λ α,β (cid:54) || u || H ∗ (Ω) (cid:82) Ω p η u dx dy for any u ∈ H ∗ (Ω) \ { } and η ∈ N α,β , passing to theinfima, (13) yields Λ α,β (cid:62) λ α,β β. This completes the proof. (cid:3)
Finally, we prove that the optimal pair of problem (12) can be characterised as follows
Lemma 4.3.
For every < α < β , let ( u, η ) ∈ H ∗ (Ω) × N α,β be an optimal pair of problem (12) .Then, u and η are related as follows η ( x, y ) = χ S u ( x, y ) for a.e. ( x, y ) ∈ Ω , where χ S u is the characteristic function of a set S u ⊂ Ω such that | S u | = β − β − α | Ω | and S u = { ( x, y ) ∈ Ω : u ( x, y ) (cid:54) t } for some t > .Proof. The proof is along the line of [11, Proposition 3.3]. For the sake of completeness we shall outlinethe main ideas.
Step 1.
Let u ∈ H ∗ (Ω) be such that || u || = 1 and consider the functional I : N α,β → R I ( η ) := (cid:90) Ω η u dx dy . We prove that the infimum problem I α,β := inf η ∈ N α,β I ( η )admits a solution η = χ S u , where S u ⊂ Ω is such that | S u | = β − β − α | Ω | and satisfies one of the following(14) S u = { ( x, y ) ∈ Ω : u ( x, y ) = 0 } or { ( x, y ) ∈ Ω : u ( x, y ) < t } ⊆ S u ⊆ { ( x, y ) ∈ Ω : u ( x, y ) (cid:54) t } , where t is defined as(15) t := sup (cid:26) s > |{ ( x, y ) ∈ Ω : u ( x, y ) < s }| < β − β − α | Ω | (cid:27) . Let S u ⊂ Ω be as above, then χ S u ∈ N α,β and one obtains I α,β (cid:54) I ( χ S u ) = (cid:90) S u u dx dy. On the other hand we claim that the following inequality holds I ( η ) (cid:62) I ( χ S u ) for any η ∈ N α,β . If this is true then one immediately obtain I α,β = I ( χ S u ) and this concludes the proof of step 1.We prove the validity of the claim by considering the cases t > t = 0 separately. If t >
0, we argue as follows (cid:90) Ω u ( χ S u − η ) dx dy (16) = (cid:90) { u
We prove that if ( u, η ) is an optimal pair as in the statement of the lemma and if S u is thecorresponding set defined according to Step 1, then ( u, χ S u ) is still an optimal pair.Set S α,β := (cid:26) S ⊂ Ω : | S | = β − β − α | Ω | (cid:27) . Since { χ S : S ∈ S α,β } ⊂ N α,β , we haveΛ α,β (cid:54) inf S ∈ S α,β inf u ∈ H ∗ (Ω) \{ } || u || H ∗ (Ω) + λ α,β ( β − α ) (cid:82) Ω χ S u dx dy (cid:82) Ω u dx dy . On the other hand, letting ( u, η ) an optimal pair as in the statement of the lemma, from Step 1 wehave || u || H ∗ (Ω) + λ α,β ( β − α ) (cid:90) Ω η u dx dy (cid:62) || u || H ∗ (Ω) + λ α,β ( β − α ) (cid:90) Ω χ S u u dx dy and thereforeΛ α,β = || u || H ∗ (Ω) + λ α,β ( β − α ) (cid:82) Ω η u dx dy (cid:82) Ω u dx dy (cid:62) || u || H ∗ (Ω) + λ α,β ( β − α ) (cid:82) Ω χ S u u dx dy (cid:82) Ω u dx dy (cid:62) inf S ∈ S α,β inf u ∈ H ∗ (Ω) \{ } || u || H ∗ (Ω) + λ α,β ( β − α ) (cid:82) Ω χ S u dx dy (cid:82) Ω u dx dy . This proves that Λ α,β = inf S ∈ S α,β inf u ∈ H ∗ (Ω) \{ } || u || H ∗ (Ω) + λ α,β ( β − α ) (cid:82) Ω χ S u dx dy (cid:82) Ω u dx dy and in particular that ( u, χ S u ) is an optimal pair. Step 3.
Let ( u, χ S u ) be the optimal pair introduced in Step 2 and let t be the number t in (14)corresponding to u . Let A t = { ( x, y ) ∈ Ω : u ( x, y ) = t } . We prove that t > | A t \ S u | = 0.Suppose by contradiction that t = 0. Since u ∈ H (Ω) we can write the Euler-Lagrange equationrelated to (12) almost everywhere and we haveΛ α,β u = ∆ u + λ α,β ( β − α ) χ S u u = ∆ u a.e. in Ω . Since u satisfies the partially hinged boundary conditions this means that it must be one of the eigen-functions listed in Proposition 2.1 which is impossible since the set of zeroes of any of the eigenfunctionsof Proposition 2.1 has zero measure thus contradicting the definition of S u which forces S u to be a setof positive measure. This proves that t > ONHOMOGENEOUS PLATES 11
Suppose now by contradiction that | A t \ S u | >
0, we have that∆ u + λ α,β ( β − α ) χ S u u = Λ α,β u a.e. in Ω . Now, exploiting the fact that u is constant in A t and t >
0, we inferΛ α,β = λ α,β ( β − α ) χ S u a.e. in A t . and hence, since λ α,β ( β − α ) χ S u = 0 a.e. in A t \ S u and | A t \ S u | >
0, we obtain Λ α,β = 0 and this isabsurd.
Step 4.
We complete the proof of the lemma. First of all, we observe that by Step 3, it is notrestrictive, up to a set of zero measure, to assume that A t \ S u = ∅ in such way that A t ⊆ S u and, inturn,(17) S u = { ( x, y ) ∈ Ω : u ( x, y ) (cid:54) t } . It remains to prove that η = χ S u a.e. in Ω. Since ( u, η ) and ( u, χ S u ) are both optimal pairs we have∆ u + λ α,β ( β − α ) χ S u u = Λ α,β u a.e. in Ω , ∆ u + λ α,β ( β − α ) η u = Λ α,β u a.e. in Ω , thus implying that ( χ S u − η ) u = 0 a.e. in Ω . It is easy to check that η = χ S u a.e. in { ( x, y ) ∈ Ω : u ( x, y ) (cid:62) t } being t >
0. In order to prove that η = χ S u a.e. in { ( x, y ) ∈ Ω : u ( x, y ) < t } , we apply (16) to u , χ S u and η observing that the inequality(16) is an equality being ( u, η ) and ( u, χ S u ) both optimal pairs. In particular we have that (cid:90) { u For every 0 < α < β , the existence of an optimal pair ( u, p ) ∈ H ∗ (Ω) × P α,β follows from Lemma 4.1.If we put η := β − pβ − α by Lemma 4.2 we deduce that ( u, η ) is an optimal pair for Λ α,β = λ α,β β . Moreoverby Lemma 4.3 we also have that η = χ S u a.e. in Ω with S u = { ( x, y ) ∈ Ω : u ( x, y ) (cid:54) t } and t as in(15). Hence we conclude that p = β − η ( β − α ) = αχ S u + βχ S cu . Proof of Proposition 3.3. We prove the two statements separately. Proof of Proposition 3.3-(i). We know that the function u ( x, y ) = ϕ ( y ) sin x introduced in (7) is an eigenfunction correspondingto the least eigenvalue of (2) with p ≡ 1. Furthermore,inf u ∈ H ∗ (Ω) \{ } (cid:107) u (cid:107) H ∗ (Ω) (cid:107) u (cid:107) = (cid:107) u (cid:107) H ∗ (Ω) (cid:107) u (cid:107) = µ , . Now, by exploiting the fact that ϕ is even and increasing in (0 , (cid:96) ) and p = p ( y ) is even, we deduce that (cid:90) Ω (1 − p ( y )) u ( x, y ) dx dy = 2 (cid:90) π (cid:90) (cid:96) (1 − p ( y )) ϕ ( y ) sin x dx dy (cid:54) ϕ ( z ) (cid:90) π (cid:90) z (1 − p ( y )) sin x dx dy + 2 ϕ ( z ) (cid:90) π (cid:90) (cid:96)z (1 − p ( y )) sin x dx dy = ϕ ( z ) π (cid:90) (cid:96) (1 − p ( y )) dy = 0 , where in the last step we have exploited the fact that (cid:82) Ω p ( y ) dx dy = | Ω | , therefore (cid:82) (cid:96) p ( y ) dy = (cid:96) .Hence, (cid:90) Ω u ( x, y ) dxdy (cid:54) (cid:90) Ω p ( y ) u ( x, y ) dxdy . From the above inequality we infer µ , = (cid:107) u (cid:107) H ∗ (Ω) (cid:107) u (cid:107) (cid:62) inf u ∈ H ∗ (Ω) \{ } (cid:107) u (cid:107) H ∗ (Ω) (cid:107)√ pu (cid:107) = λ ( p ) , and the proof of the statement follows. Proof of Proposition 3.3-(ii). The idea of the proof is similar to that applied to prove statement ( i ). By exploiting the fact thatsin( π − x ) = sin( x ) and p ( π − x ) = p ( x ) for all x ∈ (0 , π ), we deduce that (cid:90) Ω (1 − p ( x )) u ( x, y ) dx dy = 2 (cid:90) (cid:96) − (cid:96) (cid:90) π (1 − p ( x )) ϕ ( y ) sin x dx dy (cid:54) ( s ) (cid:90) (cid:96) − (cid:96) (cid:90) s (1 − p ( x )) ϕ ( y ) dx dy + 2 sin ( s ) (cid:90) (cid:96) − (cid:96) (cid:90) π s (1 − p ( x )) ϕ ( y ) dx dy = 2 sin ( s ) (cid:18)(cid:90) (cid:96) − (cid:96) ϕ ( y ) dy (cid:19) (cid:32)(cid:90) π (1 − p ( x )) dx (cid:33) = 0 , where in the last step we have exploited the assumption (cid:82) Ω p ( x ) dx dy = | Ω | , hence (cid:82) π p ( x ) dx = π .From the above inequality the proof follows as for statement ( i ).5. Proof of Theorem 3.5 and Theorem 3.6 Let 0 < α < β . In this section we restrict the admissible weights to the family P α,β defined in Theorem 3.5.Clearly, (cid:82) (cid:96) p dy = (cid:96) for all p ∈ P α,β . Let m be a positive integer, we consider the following scalarproduct in H ( − (cid:96), (cid:96) ): (cid:104) ϕ, φ (cid:105) m := (cid:90) (cid:96) − (cid:96) (cid:0) ϕ (cid:48)(cid:48) φ (cid:48)(cid:48) + 2 m (1 − σ ) ϕ (cid:48) φ (cid:48) − σm ( ϕ (cid:48)(cid:48) φ + ϕφ (cid:48)(cid:48) ) + m ϕφ (cid:1) dy . For every m (cid:62) H ( − (cid:96), (cid:96) ) that we will denote by ||| φ ||| m = ( φ, φ ) m .Let u be an eigenfunction of (4), its Fourier expansion reads u ( x, y ) = + ∞ (cid:88) m =1 ϕ m ( y ) sin( mx )with ϕ m ∈ C ([ − (cid:96), (cid:96) ]) since u ∈ H (Ω) (at least). Inserting u in (4), we get that, for every m (cid:62) ϕ m satisfies the equation(19) (cid:104) ϕ, φ (cid:105) m = λ (cid:90) (cid:96) − (cid:96) p ( y ) ϕφ dy for all φ ∈ H ( − (cid:96), (cid:96) ) ONHOMOGENEOUS PLATES 13 which is the weak formulation of the problem(20) ϕ (cid:48)(cid:48)(cid:48)(cid:48) ( y ) − m ϕ (cid:48)(cid:48) ( y ) + m ϕ ( y ) = λp ( y ) ϕ ( y ) in ( − (cid:96), (cid:96) ) ϕ (cid:48)(cid:48) ( ± (cid:96) ) − σm ϕ ( ± (cid:96) ) = 0 ϕ (cid:48)(cid:48)(cid:48) ( ± (cid:96) ) − (2 − σ ) m ϕ (cid:48) ( ± (cid:96) ) = 0 . Notice that, by elliptic regularity, any solution ϕ ∈ H ( − (cid:96), (cid:96) ) of (20) , lies in H ( − (cid:96), (cid:96) ) ⊂ C ([ − (cid:96), (cid:96) ]).Hence, the boundary conditions in (20) are satisfied pointwise. Since the bilinear form (cid:104) ϕ, φ (cid:105) m iscontinuous and coercive the eigenvalues of problem (19) may be ordered in an increasing sequence ofstrictly positive numbers diverging to + ∞ and the corresponding eigenfunctions form a complete systemin H ( − (cid:96), (cid:96) ). Whence, for what remarked so far, when p = p ( y ) there is a one to one correspondencebetween eigenvalues of (19) and eigenvalues of (4). In particular, if we denote by λ ( p ) the firsteigenvalue of (4) and by λ ( p, m ) the first eigenvalue of (19) with m (cid:62) λ ( p ) := inf u ∈ H ∗ (Ω) \{ } (cid:107) u (cid:107) H ∗ (Ω) (cid:107)√ pu (cid:107) and λ ( p, m ) := inf ϕ ∈ H ( − (cid:96),(cid:96) ) \{ } ||| ϕ ||| m (cid:107)√ pϕ (cid:107) , it is natural to conjecture that λ ( p ) = min m (cid:62) (cid:8) λ ( p, m ) (cid:9) = λ ( p, . Unfortunately, for p ∈ P α,β fixed, due to the negative terms in the norm ||| ·||| m , the monotonicity of m (cid:55)→ λ ( p, m ) is not easy to detect and we do not have a proof of the above equality for general p ; inSection 6 we give some suggestions through numerical experiments. Nevertheless, we have the followingpartial result Lemma 5.1. If p ∈ P α,β then λ ( p, m ) (cid:54) µ m, < m , where the µ m, are the numbers defined in Proposition 2.1-(i).If furthermore β (cid:54) (1 − σ )2 , then (21) λ ( p, m ) (cid:62) λ ( p, for all m (cid:62) . Proof. Let ϕ m ( y ) := (cid:2) µ / m, − (1 − σ ) m (cid:3) cosh (cid:16) y (cid:113) m + µ / m, (cid:17) cosh (cid:16) (cid:96) (cid:113) m + µ / m, (cid:17) + (cid:2) µ / m, + (1 − σ ) m (cid:3) cosh (cid:16) y (cid:113) m − µ / m, (cid:17) cosh (cid:16) (cid:96) (cid:113) m − µ / m, (cid:17) , From Proposition 2.1 it is readily deduced that ϕ m ( y ) is an eigenfunction corresponding to the leasteigenvalue of (19) with p ≡ m (cid:62) ϕ ∈ H ( − (cid:96),(cid:96) ) \{ } ||| ϕ ||| m (cid:107) ϕ (cid:107) = ||| ϕ m ||| m (cid:107) ϕ m (cid:107) = µ m, . Now, by exploiting the fact that ϕ m is even and increasing in (0 , (cid:96) ), the first part of the proof followswith the same argument of Proposition 3.3-(i), hence we omit it.Next we turn to the second estimate. Let ϕ m,p ( y ) be an eigenfunction corresponding to the leasteigenvalue of (19), with m (cid:62) p ∈ P α,β satisfying the assumption of Lemma 5.1. Inparticular, ϕ m, = ϕ m , with ϕ m as given above. Since p ( y ) (cid:54) (1 − σ )2 for every y ∈ ( − (cid:96), (cid:96) ), we get λ ( p, m ) = ||| ϕ m,p ||| m (cid:107)√ pϕ m,p (cid:107) (cid:62) − σ ) m ||| ϕ m,p ||| m (cid:107) ϕ m,p (cid:107) (cid:62) µ m, (1 − σ ) m . Then, the thesis follows by recalling that, from Proposition 2.1-(i), µ m, ∈ ((1 − σ ) m , m ) for every m (cid:62) λ ( p, < (cid:3) Hence, under the assumptions of Lemma 5.1, we have λ ( p ) = λ ( p, (cid:54) µ , = λ (1) . In particular, the weights considered in Lemma 5.1 prove to be effective in decreasing the first frequencyof (2), which is one of the main goal of the present analysis. In the following we refine the result bycarrying on a more deeper analysis. First we note that, from above, if ϕ ,p ( y ) is an eigenfunction of λ ( p, u ,p ( x, y ) := ϕ ,p ( y ) sin( x ) is an eigenfunction of λ ( p ). Therefore, ϕ ,p ( y ) and u ,p ( x, y )have the same sign.We discuss now the sign of ϕ ,p ( y ) and the simplicity of λ ( p ) in Lemma 5.2. Let m (cid:62) integer fixed, σ ∈ (0 , and let p ∈ P α,β . Then, the first eigenvalue λ ( p, m ) of problem (19) is simple and the first eigenfunction ϕ m,p ( y ) is of one sign in [ − (cid:96), (cid:96) ] .Furthermore, if the assumptions of Lemma 5.1 holds, the same conclusion holds for the first eigen-value λ ( p ) of (4), namely it is simple and the corresponding eigenfunction is given by u ,p ( x, y ) = ϕ ,p ( y ) sin( x ) , hence of one sign in Ω .Proof. We apply the decomposition with respect to dual cones technique, see [17, Chapter 3] suitablecombined with Theorem 7.1 below. We start by recalling some basic facts concerning the just mentioneddecomposition. Let H be a Hilbert space with scalar product ( ., . ) H . Let K ⊂ H be a closed nonemptycone and let K ∗ be its dual cone, namely K ∗ := { ψ ∈ H : ( ψ, φ ) H (cid:54) φ ∈ K} . Then, for any ϕ ∈ H there exists a unique ( χ, ψ ) ∈ K × K ∗ such that ϕ = χ + ψ , ( χ, ψ ) H = 0 . Now we turn to the proof of Lemma 5.2. We apply the above decomposition with H = H ( − (cid:96), (cid:96) ),( ., . ) H = (cid:104) ., . (cid:105) m and K = { ϕ ∈ H : ϕ (cid:62) − (cid:96), (cid:96) ) } . We know that λ ( p, m ) = inf ϕ ∈ H ( − (cid:96),(cid:96) ) \{ } ||| ϕ ||| m (cid:107)√ pϕ (cid:107) = ||| ϕ m,p ||| m (cid:107)√ pϕ m,p (cid:107) . For contradiction, assume that ϕ m,p changes sign. Then, we may decompose ϕ m,p = χ m,p + ψ m,p with χ m,p ∈ K \ { } and ψ m,p ∈ K ∗ \ { } .In the remaining part of this proof we need some results on a positivity preserving property which istreated in Section 7.¿From Corollary 7.2, we deduce that ψ m,p < − (cid:96), (cid:96) ). Then, replacing ϕ m,p with χ m,p − ψ m,p ,exploiting the fact that χ m,p − ψ m,p > ϕ m,p in ( − (cid:96), (cid:96) ) and the orthogonality of χ m,p and ψ m,p in H ( − (cid:96), (cid:96) ), we infer ||| χ m,p − ψ m,p ||| m (cid:107)√ p ( χ m,p − ψ m,p ) (cid:107) < ||| ϕ m,p ||| m (cid:107)√ pϕ m,p (cid:107) , a contradiction. Hence ϕ m,p (cid:62) − (cid:96), (cid:96) ) and since ϕ m,p solves (19), by Theorem 7.1 with f = λ ( p, m ) p ( y ) ϕ m,p , we conclude that ϕ m,p > − (cid:96), (cid:96) ].As concerns the simplicity, it follows by noting that if ϕ m,p and ¯ ϕ m,p are two linearly independentpositive minimizers, then ϕ m,p + t ¯ ϕ m,p is a sign-changing minimizer for some t < (cid:3) Next we focus on the sign of ϕ (cid:48) ,p ( y ) and we prove Lemma 5.3. Let σ ∈ (0 , . If p ∈ P α,β is such that β < /µ , and if ϕ ,p is a positive eigenfunctionof (19) with m = 1 corresponding to the first eigenvalue λ ( p, , then ϕ ,p is increasing in (0 , (cid:96) ) .Proof. For shortness we will write ϕ instead of ϕ ,p . Since p is even, being ϕ positive, we infer thatit is an even function. Hence, since ϕ ∈ C ([ − (cid:96), (cid:96) ]) it satisfies ϕ (cid:48) (0) = 0 = ϕ (cid:48)(cid:48)(cid:48) (0). ONHOMOGENEOUS PLATES 15 If p is continuous, then ϕ ∈ C ([ − (cid:96), (cid:96) ]) and it satisfies the equation in (20) pointwise. We recallthat the boundary conditions in (20) are satisfied pointwise also when p is not continuous. Since ϕ ispositive, β < /µ , and, by Lemma 5.1, we know that λ ( p, (cid:54) µ , , from the equation we infer(22) ϕ (cid:48)(cid:48)(cid:48)(cid:48) ( y ) − ϕ (cid:48)(cid:48) ( y ) = ( λ ( p, p ( y ) − ϕ ( y ) (cid:54) ( µ , p ( y ) − ϕ ( y ) < − (cid:96), (cid:96) )If p is not continuous, since only a finite number of points of jump discontinuity are allowed in ( − (cid:96), (cid:96) ),say { τ j } rj =1 for some integer r , the above inequality holds in each interval ( τ j , τ j +1 ). Furthermore,for any j = 1 , ..., r , the right and left fourth order derivative at τ j exists and they are given by( ϕ ) (cid:48)(cid:48)(cid:48)(cid:48)± ( τ j ) = lim y → τ ± j ϕ (cid:48)(cid:48)(cid:48)(cid:48) ( y ).First we show that(23) ϕ (cid:48) never vanishes in (0 , (cid:96) ) . By contradiction, let y ∈ (0 , (cid:96) ) be such that ϕ (cid:48) ( y ) = 0. Since ϕ (cid:48) (0) = 0 and ϕ ∈ C ([ − (cid:96), (cid:96) ]), thereexists y ∈ (0 , y ) such that ϕ (cid:48)(cid:48) ( y ) = 0 and, by (22), ( ϕ ) (cid:48)(cid:48)(cid:48)(cid:48) + ( y ) < 0. Next the following two cases mayoccur. • CASE 1: ϕ (cid:48)(cid:48)(cid:48) ( y ) (cid:54) 0. From above, ϕ (cid:48)(cid:48)(cid:48) is negative and, in turn, also ϕ (cid:48)(cid:48) is negative in a rightneighborhood of y . Since the boundary conditions in (20) yield ϕ (cid:48)(cid:48) ( (cid:96) ) = σϕ ( (cid:96) ) > 0, we infer thatthere exists y > y such that ϕ (cid:48)(cid:48) ( y ) = 0, ϕ (cid:48)(cid:48)(cid:48) ( y ) (cid:62) ϕ (cid:48)(cid:48) ( y ) (cid:54) y , y ). Whence, by (22), ϕ (cid:48)(cid:48)(cid:48)(cid:48) ( y ) < y , y ) or in each of the subintervals ( τ j , τ j +1 ) contained in ( y , y ). Since ϕ (cid:48)(cid:48)(cid:48) is continuousin [ y , y ], in any case, we have that it is strictly decreasing in [ y , y ], hence ϕ (cid:48)(cid:48)(cid:48) ( y ) < y , y ] incontradiction with ϕ (cid:48)(cid:48)(cid:48) ( y ) (cid:62) • CASE 2: ϕ (cid:48)(cid:48)(cid:48) ( y ) > 0. We distinguish two further cases.CASE 2a: ϕ (cid:48)(cid:48) (0) (cid:54) 0. By (22), ( ϕ ) (cid:48)(cid:48)(cid:48)(cid:48) + (0) < 0, hence ϕ (cid:48)(cid:48)(cid:48) ( y ) < ϕ (cid:48)(cid:48)(cid:48) ( y ) > 0, there exists y ∈ (0 , y ) such that ϕ (cid:48)(cid:48)(cid:48) ( y ) < , y ) and ϕ (cid:48)(cid:48)(cid:48) ( y ) = 0. In turn, ϕ (cid:48)(cid:48) < , y ) and by (22) ϕ (cid:48)(cid:48)(cid:48)(cid:48) ( y ) < , y ) (or in each of the subintervals ( τ j , τ j +1 ) contained in( y , y )). Since ϕ (cid:48)(cid:48)(cid:48) is continuous this lead that it is strictly decreasing in [0 , y ]. Since ϕ (cid:48)(cid:48)(cid:48) (0) = 0, weinfer ϕ (cid:48)(cid:48)(cid:48) ( y ) < 0, a contradiction.CASE 2b: ϕ (cid:48)(cid:48) (0) > 0. From ϕ (cid:48)(cid:48)(cid:48) ( y ) > ϕ (cid:48)(cid:48) ( y ) = 0 we infer that ϕ (cid:48)(cid:48) is negative in a leftneighborhood of y . Then, since ϕ (cid:48)(cid:48) (0) > 0, there exists y ∈ (0 , y ) such that ϕ (cid:48)(cid:48) ( y ) > , y ) and ϕ (cid:48)(cid:48) ( y ) = 0. Consecutively, recalling that ϕ (cid:48)(cid:48) ( y ) = 0, there exists y ∈ ( y , y ) such that ϕ (cid:48)(cid:48)(cid:48) ( y ) = 0and, by (22), we infer that ϕ (cid:48)(cid:48)(cid:48) ( y ) < y , y ), in contradiction with ϕ (cid:48)(cid:48)(cid:48) ( y ) > ϕ (cid:48) ( y ) < , (cid:96) ) or ϕ (cid:48) ( y ) > , (cid:96) ).Assume that ϕ (cid:48) ( y ) < , (cid:96) ), then ϕ (cid:48)(cid:48) (0) (cid:54) 0. Indeed, if ϕ (cid:48)(cid:48) (0) > 0, since ϕ (cid:48) (0) = 0, then ϕ (cid:48) is positive in a right neighborhood of 0, a contradiction. From ϕ (cid:48)(cid:48) (0) (cid:54) 0, together with (22) and ϕ (cid:48)(cid:48)(cid:48) (0) = 0, it follows that ϕ (cid:48)(cid:48)(cid:48) is negative in a right neighborhood of 0 and, in turn, also ϕ (cid:48)(cid:48) is negativein a right neighborhood of 0. Since, from the boundary conditions ϕ (cid:48)(cid:48) ( (cid:96) ) = σϕ ( (cid:96) ) > 0, we deduce thatthere exists y ∈ (0 , (cid:96) ) such that ϕ (cid:48)(cid:48) ( y ) = 0, ϕ (cid:48)(cid:48)(cid:48) ( y ) (cid:62) ϕ (cid:48)(cid:48) ( y ) (cid:54) , y ) . But then, from (22), ϕ (cid:48)(cid:48)(cid:48) is strictly decreasing in [0 , y ] and, recalling that ϕ (cid:48)(cid:48)(cid:48) (0) = 0 we reach a contradiction. (cid:3) All the above statements yield the proof of Theorem 3.5. Proof of Theorem 3.5 completed. The key point is to note that, by Lemma 5.3, we have(24) p ∈ P α,β ⇒ ϕ ,p increasing in (0 , (cid:96) ) . Indeed, since by (24) ϕ ,p is increasing in (0 , (cid:96) ), to prove ( i ) we may argue as in the proof of the firstpart of Lemma 5.1 with ϕ ,p instead of ϕ m . In particular, we readily infer that λ ( p , (cid:54) λ ( p , λ ( p, 1) = λ ( p ) for all p ∈ P α,β , the proof of ( i ) follows. Next we prove ( ii ). Set y := (cid:96) ( β − β − α , for every p ∈ P α,β there holds p ( y ) (cid:62) p ( y ) in [0 , y ] and p ( y ) (cid:54) p ( y ) in [ y, (cid:96) ) . Then, we may argue again as in the proof of the first part of Lemma 5.1 with ϕ ,p instead of ϕ m andconclude λ ( p, (cid:62) λ ( p, . Once more, from Lemma 5.1, λ ( p, 1) = λ ( p ) for all p ∈ P α,β and the statement of Theorem 3.5follows. Proof of Theorem 3.6. The proof readily follows by combining the statements of Lemma 5.2 and Lemma 5.3.6. Numerical Results In this section, for any m (cid:62) 1, we compute numerically the first eigenvalue λ ( p, m ) of problem (20)when p is as defined in (11). More precisely, we take p α,β ( y ) = (cid:40) β y ∈ ( − (cid:96), − y ) ∪ ( y, (cid:96) ) α y ∈ ( − y, y )where β > > α > y = (cid:96) ( β − β − α , so that (cid:82) (cid:96) pdy = (cid:96) .In terms of engineering applications, this means that we are dealing with a weight given by the pairingof two materials having different rigidities α and β , properly placed on rectangular strips, having thelength of the whole plate.Note that, since p α,β ( y ) is an even function, to determine all eigenvalues of (20), we may focus oneven and odd eigenfunctions. Indeed, if ϕ ( y ) is an eigenfunction which is neither odd or even, it isreadily verified that also ϕ ev ( y ) := ϕ ( y )+ ϕ ( − y )2 and ϕ od ( y ) := ϕ ( y ) − ϕ ( − y )2 are eigenfunctions, respectivelyeven and odd, corresponding to the same eigenvalue of ϕ ( y ). On the other hand, since by Lemma5.2, the first eigenvalue of (20) is simple and the corresponding eigenfunctions is of one sign in [ − (cid:96), (cid:96) ],we infer that it must be an even function, whence to compute λ ( p, m ) we may concentrate on eveneigenfunctions that we named ϕ ev . For any m (cid:62) ϕ ev ( y ) = h ( − y ) on [ − (cid:96), − y ] h ( y ) on ( − y, y ) h ( y ) on [ y, (cid:96) ]where h and h satisfy:(26) h (cid:48)(cid:48)(cid:48)(cid:48) ( y ) − m h (cid:48)(cid:48) ( y ) + m h ( y ) = λβh ( y ) on ( y, (cid:96) ) h (cid:48)(cid:48)(cid:48)(cid:48) ( y ) − m h (cid:48)(cid:48) ( y ) + m h ( y ) = λαh ( y ) on [0 , y ) h (cid:48)(cid:48) ( (cid:96) ) − σm h ( (cid:96) ) = 0 , h (cid:48)(cid:48)(cid:48) ( (cid:96) ) − (2 − σ ) m h (cid:48) ( (cid:96) ) = 0 ,h (cid:48) (0) = 0 , h (cid:48)(cid:48)(cid:48) (0) = 0 ,h ( y ) = h ( y ) , h (cid:48) ( y ) = h (cid:48) ( y ) ,h (cid:48)(cid:48) ( y ) = h (cid:48)(cid:48) ( y ) , h (cid:48)(cid:48)(cid:48) ( y ) = h (cid:48)(cid:48)(cid:48) ( y ) . Note that the compatibility conditions between the functions h and h , ensure that ϕ ev ∈ C ([ − (cid:96), (cid:96) ]),while h (cid:48) (0) = h (cid:48)(cid:48)(cid:48) (0) = 0 come from ϕ ev ( − y ) = ϕ ev ( y ) and its regularity. Clearly, the analyticalexpression of h ( y ) and h ( y ) depends on the roots of the characteristic polynomials related to the firsttwo equations in (26); we denote them respectively by ζ and ζ and we find that they satisfy ζ = m ± (cid:112) λβ ζ = m ± √ λα. ONHOMOGENEOUS PLATES 17 Therefore, the sign of m − √ λβ and m − √ λα determines different kinds of solutions. We introducethe following notations η α := (cid:113) m + √ λα, η β := (cid:113) m + (cid:112) λβ, ω α := (cid:113) | m − √ λα | , ω β := (cid:113) | m − (cid:112) λβ | , and we distinguish five cases: a) m > λβ > λα , implying λ < m /β and h ( y ) = a cosh (cid:0) η β y (cid:1) + b sinh (cid:0) η β y (cid:1) + c cosh (cid:0) ω β y (cid:1) + d sinh (cid:0) ω β y (cid:1) ,h ( y ) = a cosh (cid:0) η α y (cid:1) + c cosh (cid:0) ω α y (cid:1) , b) m = λβ , so that η α = m (cid:113) (cid:112) α/β , ω α = m (cid:113) − (cid:112) α/β and h ( y ) = a cosh (cid:0) √ my (cid:1) + b sinh (cid:0) √ my (cid:1) + c y + d ,h ( y ) = a cosh (cid:0) η α y (cid:1) + c cosh (cid:0) ω α y (cid:1) , c) λα < m < λβ , implying m /β < λ < m /α and h ( y ) = a cosh (cid:0) η β y (cid:1) + b sinh (cid:0) η β y (cid:1) + c cos (cid:0) ω β y (cid:1) + d sin (cid:0) ω β y (cid:1) ,h ( y ) = a cosh (cid:0) η α y (cid:1) + c cosh (cid:0) ω α y (cid:1) , d) m = λα , so that η β = m (cid:113) (cid:112) β/α , ω β = m (cid:113)(cid:112) β/α − h ( y ) = a cosh (cid:0) η β y (cid:1) + b sinh (cid:0) η β y (cid:1) + c cos (cid:0) ω β y (cid:1) + d sin (cid:0) ω β y (cid:1) ,h ( y ) = a cosh (cid:0) √ my (cid:1) + c , e) m < λα < λβ , implying λ > m /α and h ( y ) = a cosh (cid:0) η β y (cid:1) + b sinh (cid:0) η β y (cid:1) + c cos (cid:0) ω β y (cid:1) + d sin (cid:0) ω β y (cid:1) ,h ( y ) = a cosh (cid:0) η α y (cid:1) + c cos (cid:0) ω α y (cid:1) . The six coefficients involved in the definition of h and h can be determined, in each of the five cases,by imposing the boundary and compatibility conditions. We present here only case c) , since the otherscases can be treated similarly.First of all we assume that h satisfies the boundary conditions, i.e.( BCs ) (cid:40) h (cid:48)(cid:48) ( (cid:96) ) − σm h ( (cid:96) ) = 0 h (cid:48)(cid:48)(cid:48) ( (cid:96) ) − (2 − σ ) m h (cid:48) ( (cid:96) ) = 0 ⇒ ( η β − σm )[ a cosh( η β (cid:96) ) + b sinh( η β (cid:96) )]+ − ( ω β + σm )[ c cos( ω β (cid:96) ) + d sin( η β (cid:96) )] = 0( η β + ( σ − m ) η β [ a sinh( η β (cid:96) ) + b cosh( η β (cid:96) )]+( ω β − ( σ − m ) ω β [ c sin( ω β (cid:96) ) − d cos( ω β (cid:96) )] = 0 , then we impose the compatibility conditions, i.e. i ) ii ) iii ) iv ) h ( y ) = h ( y ) h (cid:48) ( y ) = h (cid:48) ( y ) h (cid:48)(cid:48) ( y ) = h (cid:48)(cid:48) ( y ) h (cid:48)(cid:48)(cid:48) ( y ) = h (cid:48)(cid:48)(cid:48) ( y ) ⇒ a cosh (cid:0) η β y (cid:1) + b sinh (cid:0) η β y (cid:1) + c cos (cid:0) ω β y (cid:1) + d sin (cid:0) ω β y (cid:1) + − a cosh (cid:0) η α y (cid:1) − c cosh (cid:0) ω α y (cid:1) = 0 a η β sinh (cid:0) η β y (cid:1) + b η β cosh (cid:0) η β y (cid:1) − c ω β sin (cid:0) ω β y (cid:1) + d ω β cos (cid:0) ω β y (cid:1) + − a η α sinh (cid:0) η α y (cid:1) − c ω α sinh (cid:0) ω α y (cid:1) = 0 a η β cosh (cid:0) η β y (cid:1) + b η β sinh (cid:0) η β y (cid:1) − c ω β cos (cid:0) ω β y (cid:1) − d ω β sin (cid:0) ω β y (cid:1) + − a η α cosh (cid:0) η α y (cid:1) − c ω α cosh (cid:0) ω α y (cid:1) = 0 a η β sinh (cid:0) η β y (cid:1) + b η β cosh (cid:0) η β y (cid:1) + c ω β sin (cid:0) ω β y (cid:1) − d ω β cos (cid:0) ω β y (cid:1) + − a η α sinh (cid:0) η α y (cid:1) − c ω α sinh (cid:0) ω α y (cid:1) = 0 . We should solve a system of six equations and six unknowns; through some algebraic manipulations,we reduce it to a system of four equations and four unknowns v = ( a , b , c , d ) T . More precisely, weget (27) ( BCs )[ η α ( h ( y ) − h ( y )) − ( h (cid:48)(cid:48) ( y ) − h (cid:48)(cid:48) ( y ))] ω α sinh( ω α y ) = [ η α ( h (cid:48) ( y ) − h (cid:48) ( y )) − ( h (cid:48)(cid:48)(cid:48) ( y ) − h (cid:48)(cid:48)(cid:48) ( y ))] cosh( ω α y )[ ω α ( h ( y ) − h ( y )) − ( h (cid:48)(cid:48) ( y ) − h (cid:48)(cid:48) ( y ))] η α sinh( η α y ) = [ ω α ( h (cid:48) ( y ) − h (cid:48) ( y )) − ( h (cid:48)(cid:48)(cid:48) ( y ) − h (cid:48)(cid:48)(cid:48) ( y ))] cosh( η α y ) . To system (27) we associate a square matrix depending on the eigenvalues M ( λ ) ∈ M ( R ), hence (27)rewrites M ( λ ) v = ; since we are interested in not trivial solutions we end up with the equation(28) f ( λ ) := det M ( λ ) = 0 with λ > . In this way, for any m (cid:62) f ( λ ) in the interval m /β < λ < m /α , ifthey exist, are the eigenvalues corresponding to eigenfunctions ϕ ev as in (25) with h and h as in c) .This procedure can be applied to each of the five cases a) − e) .The computation by hand of (28) is very involved, thus we perform it numerically in all the five caseslisted above. Our experiments reveal that cases b) and d) do not occur for 1 (cid:54) m (cid:54) M , for a suitable M which, varying α and β , always satisfies M ≈ /(cid:96) . This implies large M for small (cid:96) , as common inplates for bridges. Therefore, we focus on cases a) - c) - e) . Figure 3. Plot of f ( λ ) in the cases a) (dashed), c) and e). Here λ evm,k := λ evk ( p α,β , m ).We point out that the plot of f ( λ ) we get, see Figure 3, is qualitatively the same for each 1 (cid:54) m (cid:54) M and for all 0 < α < β taken. As Figure 3 shows: we do not find eigenvalues in case a) , since f ( λ ) > λ ∈ (0 , m /β ); the first eigenvalue λ ev ( p α,β , m ) falls always in case c) ; all the other eigenvaluescorresponding to even functions fall in case e) . Furthermore, our numerical results yield the followingbounds on eigenvalues corresponding to even eigenfunctions: m β < λ ev ( p α,β , m ) = λ ( p α,β , m ) < m , λ evk ( p α,β , m ) > m α for k (cid:62) . We are now interested in checking if (21) holds when the assumptions of Lemma 5.1 are not satisfied,i.e. if λ ev ( p α,β , m ) (cid:62) λ ev ( p α,β , 1) for m (cid:62) β (cid:29) (1 − σ )2 . To this aim we study the behaviour of the maps β (cid:55)→ λ ev ( p α,β , m ) and m (cid:55)→ λ ev ( p α,β , m ). In Figure 4 we plot some points of the map β (cid:55)→ λ ev ( p α,β , 1) for α = 0 . 5, we register asimilar behaviour for λ ev ( p α,β , m ) with m (cid:62) 2. On the other hand, in Table 1 we put the values of λ ev ( p α,β , m ) for m = 1 , . . . , 10, computed taken α = β = 1, i.e. p ≡ 1, and for two suitable choices of α and β with β satisfying or not satisfying the smallness assumption of Lemma 5.1. ONHOMOGENEOUS PLATES 19 Figure 4. Plot of β (cid:55)→ λ ev ( p α,β , 1) with (cid:96) = π ( α = 0 . Case λ ev , λ ev , λ ev , λ ev , λ ev , λ ev , λ ev , λ ev , λ ev , λ ev , α = β = 1 α = 0 . β = 1 . α = 0 . β = 20 Table 1. The eigenvalues λ evm, := λ ev ( p α,β , m ) with m = 1 , . . . , 10 and (cid:96) = π .All the numerical experiments performed suggest thatthe map β (cid:55)→ λ ev ( p α,β , m ) is decreasing and λ ev ( p α,β , m ) (cid:62) ( m − for all β > (cid:96) and α . In particular, the above lower bound for λ ev ( p α,β , m )does not depend on β and, jointly with the fact that λ ev ( p α,β , m ) < m , supports the conjecture thatthe map m (cid:55)→ λ ev ( p α,β , m ) is increasingfor any β > 1, hence the assumption β (cid:29) (1 − σ )2 of Lemma 5.1 seems a merely technical condition.7. A positivity preserving property and proof Theorem 3.7 In this section we state and prove some results about a positivity preserving property for the fourthorder differential operator(29) L m ϕ = ϕ (cid:48)(cid:48)(cid:48)(cid:48) − m ϕ (cid:48)(cid:48) + m ϕ , m ∈ N , m (cid:62) , ϕ : [ − (cid:96), (cid:96) ] → R , subject to the boundary conditions introduced in (20). Theorem 7.1. Let m (cid:62) be an integer, σ ∈ (0 , and let f ∈ L ( − (cid:96), (cid:96) ) . Furthermore, assume that w m ∈ H ( − (cid:96), (cid:96) ) is a weak solution to the problem (30) w (cid:48)(cid:48)(cid:48)(cid:48) ( y ) − m w (cid:48)(cid:48) ( y ) + m w ( y ) = f ( y ) y ∈ ( − (cid:96), (cid:96) ) w (cid:48)(cid:48) ( ± (cid:96) ) − σm w ( ± (cid:96) ) = 0 w (cid:48)(cid:48)(cid:48) ( ± (cid:96) ) − (2 − σ ) m w (cid:48) ( ± (cid:96) ) = 0 . namely (31) (cid:104) w, φ (cid:105) m = (cid:90) (cid:96) − (cid:96) f φ for all φ ∈ H ( − (cid:96), (cid:96) ) . Then the following implication holds f (cid:62) in ( − (cid:96), (cid:96) ) ( f (cid:54)≡ ⇒ w ( y ) > in [ − (cid:96), (cid:96) ] . Hence, the operator L m defined in (29) , under the boundary conditions in (30) , satisfies the positivitypreserving property. As a consequence of Theorem 7.1 we have Corollary 7.2. Let m (cid:62) and < σ < . Furthermore, set K := { φ ∈ H ( − (cid:96), (cid:96) ) : φ (cid:62) in ( − (cid:96), (cid:96) ) } and assume that w ∈ H ( − (cid:96), (cid:96) ) satisfies (32) (cid:104) w, φ (cid:105) m (cid:54) for all φ ∈ K . Then either w ≡ or w < in ( − (cid:96), (cid:96) ) . Proof. Let f ∈ K and let φ f be the unique solution to (cid:104) φ f , ψ (cid:105) m = (cid:90) (cid:96) − (cid:96) f ψ dy for all ψ ∈ H ( − (cid:96), (cid:96) ) . By Theorem 7.1, φ f ∈ K . Inserting φ f in (32) we infer (cid:90) (cid:96) − (cid:96) f w dy = (cid:104) w, φ f (cid:105) m (cid:54) f ∈ K . Hence, w (cid:54) − (cid:96), (cid:96) ). By contradiction, assume that w (cid:54) < − (cid:96), (cid:96) ). Then, if Z := { y ∈ ( − (cid:96), (cid:96) ) : w ( y ) = 0 } , we have that the characteristic function of Z satisfies χ Z (cid:62) χ Z (cid:54)≡ 0. Let now φ ∈ H ( − (cid:96), (cid:96) ) satisfy (cid:104) φ , ψ (cid:105) m = (cid:90) (cid:96) − (cid:96) χ Z ψ dy for all ψ ∈ H ( − (cid:96), (cid:96) ) . Since, by elliptic regularity, φ ∈ C ([ − (cid:96), (cid:96) ]) and, by Theorem 7.1, φ > − (cid:96), (cid:96) ], we deduce that forevery φ ∈ H ( − (cid:96), (cid:96) ) there exist t (cid:54) (cid:54) t : φ + t φ (cid:62) φ + t φ (cid:54) − (cid:96), (cid:96) ]. Furthermore, bydefinition of φ we have (cid:104) φ , w (cid:105) m = (cid:90) (cid:96) − (cid:96) χ Z w dy = 0 . Combining this with (32), we deduce0 (cid:62) (cid:104) φ + t φ , w (cid:105) m = (cid:104) φ, w (cid:105) m and 0 (cid:54) (cid:104) φ + t φ , w (cid:105) m = (cid:104) φ, w (cid:105) m . Namely, (cid:104) φ, w (cid:105) m = 0 for all φ ∈ H ( − (cid:96), (cid:96) ) . Taking φ = w in the above inequality we conclude w ≡ − (cid:96), (cid:96) ) and the proof follows. (cid:3) We conclude this section with the proof of Theorem 7.1. Proof of Theorem 7.1. The proof follows by a direct inspection of the sign of the unique solution to (31). First we note that,for m (cid:62) f ∈ L ( − (cid:96), (cid:96) ), all solutions to the equation w (cid:48)(cid:48)(cid:48)(cid:48) ( y ) − m w (cid:48)(cid:48) ( y ) + m w ( y ) = f in D (cid:48) ( R ) , where f denotes the trivial extension of f to R , write w ( y ) = c cosh( my ) + c sinh( my ) + c y cosh( my ) + c y sinh( my ) + w p ( y ) , with c , c , c , c ∈ R and w p ( y ) = ( q m ∗ f )( y ) = (cid:90) + ∞−∞ q m ( t ) f ( y − t ) dt where q m ( y ) = (1 + m | y | ) e − m | y | m . ONHOMOGENEOUS PLATES 21 Exploiting the regularity of q m , it follows that all the above solutions belong to C ( R ) (the regularitycan be improved by increasing the regularity of f ); the thesis can be reached proving that(33) (cid:101) w ( y ) = c cosh( my ) + c sinh( my ) + c y cosh( my ) + c y sinh( my ) > w p ( y ) (cid:62) c , c , c , c ∈ R in such a way that: (cid:40) w (cid:48)(cid:48) ( ± (cid:96) ) − σm w ( ± (cid:96) ) = 0 w (cid:48)(cid:48)(cid:48) ( ± (cid:96) ) − (2 − σ ) m w (cid:48) ( ± (cid:96) ) = 0 , then the restriction of w to [ − (cid:96), (cid:96) ], that we will still denote with w , is the unique solution to (31). Moreprecisely, by imposing the above conditions we obtain the system ( c m + 2 c m ) cosh( m(cid:96) ) + ( c m + 2 c m ) sinh( m(cid:96) ) + c m (cid:96) cosh( m(cid:96) ) + c m (cid:96) sinh( m(cid:96) ) + w (cid:48)(cid:48) p ( (cid:96) ) = σm [ c cosh( m(cid:96) ) + c sinh( m(cid:96) ) + c (cid:96) cosh( m(cid:96) ) + c (cid:96) sinh( m(cid:96) ) + w p ( (cid:96) )]( c m + 2 c m ) cosh( m(cid:96) ) − ( c m + 2 c m ) sinh( m(cid:96) ) − c m (cid:96) cosh( m(cid:96) ) + c m (cid:96) sinh( m(cid:96) ) + w (cid:48)(cid:48) p ( − (cid:96) ) = σm [ c cosh( m(cid:96) ) − c sinh( m(cid:96) ) − c (cid:96) cosh( m(cid:96) ) + c (cid:96) sinh( m(cid:96) ) + w p ( − (cid:96) )]( c m + 3 c m ) cosh( m(cid:96) ) + ( c m + 3 c ) sinh( m(cid:96) ) + c m (cid:96) cosh( m(cid:96) ) + c m (cid:96) sinh( m(cid:96) ) + w (cid:48)(cid:48)(cid:48) p ( (cid:96) ) = − ( σ − m [( c m + c ) cosh( m(cid:96) ) + ( c m + c ) sinh( m(cid:96) ) + c m(cid:96) cosh( m(cid:96) ) + c m(cid:96) sinh( m(cid:96) ) + w (cid:48) p ( (cid:96) )]( c m + 3 c m ) cosh( m(cid:96) ) − ( c m + 3 c ) sinh( m(cid:96) ) − c m (cid:96) cosh( m(cid:96) ) + c m (cid:96) sinh( m(cid:96) ) + w (cid:48)(cid:48)(cid:48) p ( − (cid:96) ) = − ( σ − m [( c m + c ) cosh( m(cid:96) ) − ( c m + c ) sinh( m(cid:96) ) − c m(cid:96) cosh( m(cid:96) ) + c m(cid:96) sinh( m(cid:96) ) + w (cid:48) p ( − (cid:96) )]which decouples in the following two systems c [2 m (1 − σ ) cosh( m(cid:96) )] + c [4 m cosh( m(cid:96) ) + 2 m (1 − σ ) (cid:96) sinh( m(cid:96) )] = σm [ w p ( (cid:96) ) + w p ( − (cid:96) )] − [ w (cid:48)(cid:48) p ( (cid:96) ) + w (cid:48)(cid:48) p ( − (cid:96) )] c [2 m ( σ − 1) sinh( m(cid:96) )] + c [2 m ( σ + 1) sinh( m(cid:96) ) + 2 m ( σ − (cid:96) cosh( m(cid:96) )] = − ( σ − m [ w (cid:48) p ( (cid:96) ) − w (cid:48) p ( − (cid:96) )] − [ w (cid:48)(cid:48)(cid:48) p ( (cid:96) ) − w (cid:48)(cid:48)(cid:48) p ( − (cid:96) )] c [2 m (1 − σ ) sinh( m(cid:96) )] + c [4 m sinh( m(cid:96) ) + 2 m (1 − σ ) (cid:96) cosh( m(cid:96) )] = σm [ w p ( (cid:96) ) − w p ( − (cid:96) )] − [ w (cid:48)(cid:48) p ( (cid:96) ) − w (cid:48)(cid:48) p ( − (cid:96) )] c [2 m ( σ − 1) cosh( m(cid:96) )] + c [2 m ( σ + 1) cosh( m(cid:96) ) + 2 m ( σ − (cid:96) sinh( m(cid:96) )] = − ( σ − m [ w (cid:48) p ( (cid:96) ) + w (cid:48) p ( − (cid:96) )] − [ w (cid:48)(cid:48)(cid:48) p ( (cid:96) ) + w (cid:48)(cid:48)(cid:48) p ( − (cid:96) )] . By setting F m ( (cid:96) ) := (3 + σ ) sinh( m(cid:96) ) cosh( m(cid:96) ) − m(cid:96) (1 − σ ) > ,F m ( (cid:96) ) := (3 + σ ) sinh( m(cid:96) ) cosh( m(cid:96) ) + m(cid:96) (1 − σ ) > ,A m ( (cid:96) ) := (1 + σ ) sinh( m(cid:96) ) − (1 − σ ) m(cid:96) cosh( m(cid:96) ) , B m ( (cid:96) ) := 2 cosh( m(cid:96) ) + (1 − σ ) m(cid:96) sinh( m(cid:96) ) ,A m ( (cid:96) ) := (1 + σ ) cosh( m(cid:96) ) − (1 − σ ) m(cid:96) sinh( m(cid:96) ) , B m ( (cid:96) ) := 2 sinh( m(cid:96) ) + (1 − σ ) m(cid:96) cosh( m(cid:96) ) ,V m ( (cid:96) ) := σm w p ( (cid:96) ) − w (cid:48)(cid:48) p ( (cid:96) ) , W m ( (cid:96) ) := ( σ − m w (cid:48) p ( (cid:96) ) + w (cid:48)(cid:48)(cid:48) p ( (cid:96) ) ,V m ( − (cid:96) ) := σm w p ( − (cid:96) ) − w (cid:48)(cid:48) p ( − (cid:96) ) , W m ( − (cid:96) ) := ( σ − m w (cid:48) p ( − (cid:96) ) + w (cid:48)(cid:48)(cid:48) p ( − (cid:96) ) , the solutions to the above systems write c = mA m ( (cid:96) )[ V m ( (cid:96) ) + V m ( − (cid:96) )] + B m ( (cid:96) )[ W m ( (cid:96) ) − W m ( − (cid:96) )]2 m (1 − σ ) F m ( (cid:96) ) c = mA m ( (cid:96) )[ V m ( (cid:96) ) − V m ( − (cid:96) )] + B m ( (cid:96) )[ W m ( (cid:96) ) + W m ( − (cid:96) )]2 m (1 − σ ) F m ( (cid:96) ) c = m cosh( m(cid:96) )[ V m ( (cid:96) ) − V m ( − (cid:96) )] − sinh( m(cid:96) )[ W m ( (cid:96) ) + W m ( − (cid:96) )]2 m F m ( (cid:96) ) c = m sinh( m(cid:96) )[ V m ( (cid:96) ) + V m ( − (cid:96) )] − cosh( m(cid:96) )[ W m ( (cid:96) ) − W m ( − (cid:96) )]2 m F m ( (cid:96) ) . By exploiting the symmetry of w m , for i = 0 and i = 2, we have w ( i ) p ( (cid:96) ) = (cid:90) (cid:96) q ( i ) m ( t ) f ( (cid:96) − t ) dt , w ( i ) p ( − (cid:96) ) = (cid:90) (cid:96) q ( i ) m ( t ) f ( − (cid:96) + t ) dt , while, for i = 1 and i = 3, we have w ( i ) p ( (cid:96) ) = (cid:90) (cid:96) q ( i ) m ( t ) f ( (cid:96) − t ) dt , w ( i ) p ( − (cid:96) ) = − (cid:90) (cid:96) q ( i ) m ( t ) f ( − (cid:96) + t ) dt . Hence, V m ( (cid:96) ) = (cid:90) (cid:96) e − mt m (1 + σ − mt (1 − σ )) f ( (cid:96) − t ) dt , W m ( (cid:96) ) = (cid:90) (cid:96) e − mt mt (1 − σ )) f ( (cid:96) − t ) dt and V m ( − (cid:96) ) = (cid:90) (cid:96) e − mt m (1 + σ − mt (1 − σ )) f ( − (cid:96) + t ) dt , W m ( − (cid:96) ) = − (cid:90) (cid:96) e − mt mt (1 − σ )) f ( − (cid:96) + t ) dt . First of all we study the sign of the coefficients c and c . Since F m ( (cid:96) ) > c has the same sign of mA m ( (cid:96) )[ V m ( (cid:96) ) + V m ( − (cid:96) )] + B m ( (cid:96) )[ W m ( (cid:96) ) − W m ( − (cid:96) )] = (cid:90) (cid:96) e − mt A m ( (cid:96) )(1 + σ − m (1 − σ ) t ) + B m ( (cid:96) )(2 + m (1 − σ ) t )] [ f ( (cid:96) − t ) + f ( − (cid:96) + t )] dt (34) = [sinh( m(cid:96) )((1 + σ ) + 2 m(cid:96) (1 − σ )) + cosh( m(cid:96) )(4 − (1 − σ ) m(cid:96) )] (cid:90) (cid:96) e − mt f ( (cid:96) − t ) + f ( − (cid:96) + t )] dt +(35) + m (1 − σ )[2 cosh( m(cid:96) ) − (1 + σ ) sinh( m(cid:96) )+(1 − σ ) m(cid:96) (cosh( m(cid:96) )+sinh( m(cid:96) ))] (cid:90) (cid:96) e − mt t f ( (cid:96) − t )+ f ( − (cid:96) + t )] dt . We observe that(36) 2 cosh( z ) > (1 + σ ) sinh( z )for z > σ ∈ (0 , z (cid:55)→ g ( z ) := sinh( z )((1 + σ ) + 2 z (1 − σ )) + cosh( z )(4 − (1 − σ ) z )and we compute its derivative g (cid:48) ( z ) = 2 σ (1 + σ ) cosh( z ) + 2(3 − σ ) sinh( z ) + z (1 − σ )[2 cosh( z ) − (1 + σ ) sinh( z )] . Thanks to (36), for z > g (cid:48) ( z ) > g ( z ) is always positive ( g (0) = 4) and in particular c > c depends on m sinh( m(cid:96) )[ V m ( (cid:96) ) + V m ( − (cid:96) )] − cosh( m(cid:96) )[ W m ( (cid:96) ) − W m ( − (cid:96) )] = (cid:90) (cid:96) e − mt (cid:2) (1 + σ ) sinh( m(cid:96) ) − m(cid:96) ) − mt (1 − σ )(sinh( m(cid:96) ) + cosh( m(cid:96) )) (cid:3) [ f ( (cid:96) − t ) + f ( − (cid:96) + t )] dt that, applying again (36), gives c < σ ∈ (0 , 1) and m(cid:96) > c and c ; since the sign of c is not known a priori,we study the sign of 2 m ( | c | ± c ), i.e. (cid:90) (cid:96) e − mt F m ( (cid:96) ) (cid:2) m(cid:96) ) − (1 + σ ) sinh( m(cid:96) ) + mt (1 − σ )(sinh( m(cid:96) ) + cosh( m(cid:96) )) (cid:3) [ f ( (cid:96) − t ) + f ( − (cid:96) + t )] dt ± (cid:90) (cid:96) e − mt F m ( (cid:96) ) (cid:2) (1 + σ ) cosh( m(cid:96) ) − m(cid:96) ) − mt (1 − σ )(sinh( m(cid:96) ) + cosh( m(cid:96) )) (cid:3) [ f ( (cid:96) − t ) − f ( − (cid:96) + t )] dt . Recalling that 0 < F m ( (cid:96) ) < F m ( (cid:96) ), we obtain the positivity of m (1 − σ )[sinh( m(cid:96) )+cosh( m(cid:96) )] (cid:26)(cid:20) F m ( (cid:96) ) ∓ F m ( (cid:96) ) (cid:21)(cid:90) (cid:96) e − mt t f ( (cid:96) − t ) dt + (cid:20) F m ( (cid:96) ) ± F m ( (cid:96) ) (cid:21)(cid:90) (cid:96) e − mt t f ( − (cid:96) + t ) dt (cid:27) ;thus 2 m ( | c | ± c ) > m(cid:96) ) − (1 + σ ) sinh( m(cid:96) ) F m ( (cid:96) ) ± (1 + σ ) cosh( m(cid:96) ) − m(cid:96) ) F m ( (cid:96) ) > ONHOMOGENEOUS PLATES 23 the achievement follows from the positivity of (cid:0) cosh( z ) ∓ sinh( z ) (cid:1)(cid:0) ± (1 + σ ) (cid:1) for all z > σ ∈ (0 , m (cid:62) 1, we set (cid:101) ψ ( t ) := m (cid:101) w ( t/m ) = c m cosh t + c m sinh t + c t cosh t + c t sinh t and we focus on the qualitative behaviour of (cid:101) ψ where, from above, c > c < c < c < − c . Clearly, (cid:101) ψ ( t ) is continuous and differentiable on R , moreover (cid:101) ψ (0) = m c > (cid:101) ψ ( t ) ∼ c ± c te | t | → −∞ for t → ±∞ . This fact implies that (cid:101) ψ ( t ) has at least two zeros of opposite sign on R ; we prove now that (cid:101) ψ ( t ) has exactly twodistinct zeros on R .We know that (cid:101) ψ ( t ) = 0 if and only if α ( t ) := ( c m + c t ) tanh t + c t + c m = 0 . Computing α (cid:48) ( t ) = 12 cosh ( t ) (2 c cosh ( t ) + c sinh(2 t ) + 2 c t + 2 c m ) we observe that(37) ∃ ! t ∈ R : α (cid:48) ( t ) = 0 . This follows because β ( t ) := 2 c cosh ( t ) + c sinh(2 t ) + 2 c t + 2 c m is always decreasing on R ; indeed c < | c | > | c | so that β (cid:48) ( t ) = 2( c sinh(2 t ) + c cosh(2 t ) + c ) < 0. Moreover β ( t ) ∼ c ± c e | t | → ∓∞ for t → ±∞ .Now let us suppose for contradiction that (cid:101) ψ ( t ) has more than two zeros on R , for instance it has 3 distinct zeros t < t < t ; this implies that α ( t ) has 3 distinct zeros, then, the Rolle’s Theorem applied to α ( t ) in the intervals[ t , t ] and [ t , t ] ensures the existence of at least two points in which α (cid:48) ( t ) = 0 on R and this contradicts (37).Hence, (cid:101) ψ , and in turn also (cid:101) w , has exactly two zeros of opposite sign on R .Since (cid:101) w ( y ) has exactly two zeros of opposite sign on R and (cid:101) w (0) > 0, if we prove that (cid:101) w ( ± (cid:96) ) > (cid:101) w ( ± (cid:96) ) = c cosh( m(cid:96) ) ± c sinh( m(cid:96) ) ± c (cid:96) cosh( m(cid:96) ) + c (cid:96) sinh( m(cid:96) ), inparticular we consider2 m (cid:101) w ( (cid:96) ) = (cid:90) (cid:96) e − mt m [ C m ( (cid:96) ) f ( (cid:96) − t ) + C m ( (cid:96) ) f ( − (cid:96) + t )] dt + (cid:90) (cid:96) e − mt t D m ( (cid:96) ) f ( (cid:96) − t ) + D m ( (cid:96) ) f ( − (cid:96) + t )] dt m (cid:101) w ( − (cid:96) ) = (cid:90) (cid:96) e − mt m [ C m ( (cid:96) ) f ( (cid:96) − t ) + C m ( (cid:96) ) f ( − (cid:96) + t )] dt + (cid:90) (cid:96) e − mt t D m ( (cid:96) ) f ( (cid:96) − t ) + D m ( (cid:96) ) f ( − (cid:96) + t )] dt where C m ( (cid:96) )= 41 − σ (cid:18) cosh ( m(cid:96) ) F m ( (cid:96) ) + sinh ( m(cid:96) ) F m ( (cid:96) ) (cid:19) + (1 + σ ) − σ ) sinh(2 m(cid:96) ) (cid:18) F m ( (cid:96) ) + 1 F m ( (cid:96) ) (cid:19) − m(cid:96) (1 + σ ) (cid:18) F m ( (cid:96) ) − F m ( (cid:96) ) (cid:19) D m ( (cid:96) ) = 2 (cid:18) cosh ( m(cid:96) ) F m ( (cid:96) ) + sinh ( m(cid:96) ) F m ( (cid:96) ) (cid:19) − σ m(cid:96) ) (cid:18) F m ( (cid:96) ) + 1 F m ( (cid:96) ) (cid:19) + m(cid:96) (1 − σ ) (cid:18) F m ( (cid:96) ) − F m ( (cid:96) ) (cid:19) C m ( (cid:96) )= 41 − σ (cid:18) cosh ( m(cid:96) ) F m ( (cid:96) ) − sinh ( m(cid:96) ) F m ( (cid:96) ) (cid:19) + (1 + σ ) − σ ) sinh(2 m(cid:96) ) (cid:18) F m ( (cid:96) ) − F m ( (cid:96) ) (cid:19) − m(cid:96) (1 + σ ) (cid:18) F m ( (cid:96) ) + 1 F m ( (cid:96) ) (cid:19) D m ( (cid:96) ) = 2 (cid:18) cosh ( m(cid:96) ) F m ( (cid:96) ) − sinh ( m(cid:96) ) F m ( (cid:96) ) (cid:19) − σ m(cid:96) ) (cid:18) F m ( (cid:96) ) − F m ( (cid:96) ) (cid:19) + m(cid:96) (1 − σ ) (cid:18) F m ( (cid:96) ) + 1 F m ( (cid:96) ) (cid:19) . The final part of the proof is devoted to prove that the coefficients C m ( (cid:96) ) , D m ( (cid:96) ) , C m ( (cid:96) ) and D m ( (cid:96) ) are positive.We recall that 1 F m ( (cid:96) ) + 1 F m ( (cid:96) ) = (3 + σ ) sinh(2 m(cid:96) ) F m ( (cid:96) ) F m ( (cid:96) ) > F m ( (cid:96) ) − F m ( (cid:96) ) = 2 m(cid:96) (1 − σ ) F m ( (cid:96) ) F m ( (cid:96) ) > , and we introduce four maps related respectively to the previous coefficients z (cid:55)→ p ( z ) := 2(3 + σ )1 − σ sinh(2 z ) cosh(2 z ) + 4 z + (1 + σ ) (3 + σ )2(1 − σ ) sinh (2 z ) − − σ ) z z (cid:55)→ q ( z ) := 3 + σ z )[2 cosh(2 z ) − (1 + σ ) sinh(2 z )] + 2(1 − σ ) z + 2(1 − σ ) z z (cid:55)→ r ( z ) := 2(3 + σ )1 − σ sinh(2 z ) + z [4 cosh(2 z ) − σ ) sinh(2 z )] z (cid:55)→ s ( z ) := (3 + σ ) sinh(2 z ) + (1 − σ ) z [2 cosh(2 z ) − (1 + σ ) sinh(2 z )] + (1 − σ )(3 + σ ) z sinh(2 z ) . Thanks to (36) q ( z ), r ( z ) and s ( z ) are always positive for z > σ ∈ (0 , p ( z ), due to the following inequality(1 + σ ) (3 + σ )1 − σ sinh (2 z ) > (2 z ) > (1 − σ )(2 z ) . This completes the proof. Proof of Theorem 3.7. The proof readily follows as a corollary of Theorem 7.1 by exploiting the same separation of variables performedin the Proof of Theorem 3.5. Acknowledgments. The first three authors are members of the Gruppo Nazionale per l’Analisi Matematica,la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and arepartially supported by the INDAM-GNAMPA 2019 grant “Analisi spettrale per operatori ellittici con condizioni diSteklov o parzialmente incernierate” and by the PRIN project “Direct and inverse problems for partial differentialequations: theoretical aspects and applications” (Italy). 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E-mail address: [email protected] Indian Institute of Science Education and Research,Dr. Homi Bhabha Road, Pashan, Pune 411008, India.