Nečesal, Petr

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 11Petr NečesalThe beam operator and the Fučík spectrumIn: Marek Fila and Angela Handlovičová and Karol Mikula and Milan Medveď and Pavol Quittner and Daniel Ševčovič(eds.): Proceedings of Equadiff 11, International Conference on Differential Equations. Czecho-Slovak series,Bratislava, July 25-29, 2005, [Part 2] Minisymposia and contributed talks. Comenius University Press, Bratislava,2007. Presented in electronic form on the Internet. pp. 303--310.Persistent URL: http://dml.cz/dmlcz/700425Terms of use:© Comenius University, 2007Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documentsstrictly for personal use. Each copy of any part of this document must contain these Terms of use.This paper has been digitized, optimized for electronic delivery and stamped with digitalsignature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czProceedings of Equadiff-112005, pp. 303–310ISBN 978-80-227-2624-5THE BEAM OPERATOR AND THE FUČÍK SPECTRUM∗PETR NEČESAL†Abstract. The aim of this paper is mainly to introduce a new variational approach in order toobtain a robust and global algorithm which is suitable for the exploration of unknown Fuč́ık spectrumstructure in the case of a beam operator. Qualitative results of numerical experiments that involveunknown branches of the Fuč́ık spectrum are essential for better understanding of the behaviour of PDEmodels such as suspension bridge models.Key words. The Fuč́ık spectrum, suspension bridge, jumping nonlinearity, beam operator, varia-tional methods.AMS subject classifications. 35B10, 35L70, 65N25.1. Introduction. Let L : dom(L) ⊂ L2(Ω) → L2(Ω) be a linear self-adjoint opera-tor, Ω ⊂ RN being open and bounded set. The setΣ(L) :={(α, β) ∈ R2 : Lu = αu+ − βu− has a nontrivial solution}(1.1)is called the Fuč́ık spectrum, where u+ := max{u, 0} and u− := max{−u, 0}. This setwas firstly introduced by Fuč́ık (see [10]) in order to extend existence results for semi-linear boundary value problems. Nowadays, several recent papers such as [1], [2] or [9]deal with the structure of the Fuč́ık spectrum Σ(L) in the case of a general operator L.For instance in [1], authors give a description of the Fuč́ık spectrum of L away from itsessential spectrum, they give local and global results describing the Fuč́ık spectrum of Land also existence results for semi-linear equations using degree computations betweenthe Fuč́ık curves. All the same there are still many open problems, especially in applyingthese general results in particular cases of partial differential operators such as wave orbeam operators.The Fuč́ık spectrum plays an important role in several mathematical models. Let usmention the following one-dimensional nonlinear model of a suspension bridgeutt(x, t) + uxxxx(x, t) + bu+(x, t) = h(x, t), (x, t) ∈ (0, π)× R,u(0, t) = u (π, t) = uxx(0, t) = uxx (π, t) = 0, t ∈ R,u(x, t) = u(x, t + T ), (x, t) ∈ (0, π)× R.(1.2)This problem can be interpreted as a normalized model of the vertical motion of a sus-pension bridge (see e.g. Drábek, Holubová, Matas and Nečesal [7]). The Fuč́ık spectrumof the related beam operator indicates points of resonance for the model (1.2). But theFuč́ık spectrum for such beam operator is not explicitely known (see [8] for more carefulanalysis). Thus we design an algorithm in order to explore the Fuč́ık curves numerically.∗This work is a joint work with Gabriela Holubová and was supported by the Grant Agency of CzechRepublic, project # GA201/03/0671, and the Ministry of Education of the Czech Republic, ResearchPlan # MSM4977751301.†Centre of Applied Mathematics, Department of Mathematics, Faculty of Applied Sciences, Universityof West Bohemia, Czech Republic (pnecesal@kma.zcu.cz)303304 P. Nečesal2. The Fuč́ık spectrum for a beam operator. Let us briefly recall in this sectionwhat is known about the Fuč́ık spectrum for the following problemutt(x, t) + uxxxx(x, t) + αu+(x, t)− βu−(x, t) = 0, (x, t) ∈ (0, π)× R,u (0, t) = u (π, t) = uxx (0, t) = uxx (π, t) = 0, t ∈ R,u(x, t) = u(x, t + T ), (x, t) ∈ (0, π)× R.(2.1)Let us setω := 2π/T, Ωω := (0, π)× (0, 2π/ω) , Hω := L2(Ωω)with a standard L2-norm ‖·‖ and standard inner product 〈·, ·〉. Let us define for k ∈ Nand l ∈ Zλωk,l := l2ω2 − k4, ϕωk,l(x, t) :=2√ω/π sin kx sin lωt for k ∈ N, l ∈ N,√ω/π sin kx for k ∈ N, l = 0,2√ω/π sin kx cos lωt for k ∈ N, −l ∈ N.The set {ϕωk,l : k ∈ N, l ∈ Z} forms an orthonormal basis in Hω. The abstract realizationof the beam differential operator u 7→ −(utt + uxxxx) with the periodic and the Navierboundary conditions from (2.1) is the linear operator Lω : dom (Lω) ⊂ Hω → Hω definedbydom (Lω) :={u ∈ H :+∞∑k=1+∞∑l=−∞∣∣λωk,l∣∣2 ∣∣uωk,l∣∣2 < +∞},Lωu(x, t) :=+∞∑k=1+∞∑l=−∞λωk,l uωk,l ϕωk,l(x, t),where uωk,l := 〈u, ϕωk,l〉. Then Lω is densely defined, closed and self-adjoint operator.Moreover, Lω has a pure point spectrum σ (Lω) made of eigenvalues {λωk,l : k ∈ N, l ∈ Z}.Due to orthogonality of basis {ϕωk,l} in Hω, it is possible to locate some inadmissible areasof R2 that have a zero intersection with the Fuč́ık spectrum Σ(Lω). More precisely, theproblem (2.1) has only a trivial solution for any(α, β) ∈ (−∞, λω1,0)× (λω1,0,+∞) ∪ (λω1,0,+∞)× (−∞, λω1,0) ∪ (λi, λj)× (λi, λj),where λi and λj are two successive eigenvalues of Lω (see Fig. 2.1 and [20] for moredetails). It is convenient to form eigenvalues {λωk,l : k ∈ N, l ∈ Z} into an infinite-dimensional matrix Λ in the following wayΛ :=λω1,0 λω1,1 λω1,2 λω1,3 λω1,4 . . .λω2,0 λω2,1 λω2,2 λω2,3 λω2,4 . . .λω3,0 λω3,1 λω3,2 λω3,3 λω3,4λω4,0 λω4,1 λω4,2 λω4,3 λω4,4....... . . .If we take into account only such solutions of the problem (2.1), which are of theform u(x, t) = y(t) sinx, then the problem (2.1) can be reduced to the following problemEquadiff-11. The beam operator and the Fuč́ık spectrum 305α−256 −81 −16 35 99 255−256−81−163599255β T = π(−1,−1)Fig. 2.1. Known parts of the Fuč́ık spectrum Σ(Lω) of the beam operator.for the periodic ordinary differential operator{y′′(t) + (α + 1)y+(t)− (β + 1)y−(t) = 0, t ∈ R,y(t) = y(t + T ), t ∈ R.(2.2)The Fuč́ık spectrum for such periodic problem (2.2) is given by explicit analytic formulas(see [6]), the particular Fuč́ık curves emanate from the points on the diagonal α = βdetermined by the eigenvalues which are located in the first row of the matrix Λ (seeFig. 2.1).On the other hand, if we focus only on time independent solutions of the problem(2.1) u(x, t) = y(x), then the problem (2.1) reads as the following Navier boundary valueproblem containing the fourth order ordinary differential operator{yIV(x) + αy+(x)− βy−(x) = 0, x ∈ (0, π) ,y(0) = y′′(0) = y(π) = y′′(π) = 0.(2.3)The corresponding Fuč́ık spectrum of (2.3) is not given by explicit analytic formulas, butthe existence of particular Fuč́ık curves is proved by Krejč́ı (see [16]). In other words, eacheigenvalue in the first column of the matrix Λ gives arise to one or two Fuč́ık curves, whichcan be explored using a standard continuation method combined with an one-dimensionalshooting method (see Fig. 2.1 and [21] for details).306 P. NečesalClearly, we expect that all other eigenvalues, which are located not only in the firstrow or column of the matrix Λ, give arise to other nontrivial Fuč́ık curves. Our expectationis hardly supported by existence results for a general self-adjoint operator by Ben-Naoum,Fabry and Smets (see [1]). In our particular case of the beam operator, their results giveus the local existence of Fuč́ık curves close to the diagonal α = β in the case of simpleeigenvalues.As for the numerical methods for the partial differential Fuč́ık spectrum problems,there exists only limited group of sporadic numerical results and experiments. One wayis to combine for instance Newton’s method and Mountain Pass Algorithm to localizethe higher Fuč́ık curves (see e.g. results in [13] for the Laplace operator). Alternativeapproach is based on a continuation shooting method presented in [19] or [20], whichmeans a suitable approach how to explore the Fuč́ık curves for the wave operator u 7→−(utt − uxx). On the other hand, in the case of the beam operator Lω, a continuationshooting method is completely inapplicable due to instability of solutions of the initialboundary value problem related to (2.1) with respect to small perturbations of initialdata and parameters α and β. Thus, the goal of the following two sections is to formulatethe main ideas of our new approach that is completely different from previous methodsand that will be suitable for exploring the Fuč́ık curves including their asymptotes in thecase of the beam operator Lω.3. Variational approach. Let µ ∈ R \ σ(L), where σ(L) denotes the spectrum ofL. Then using the following transformation TµTµ = Tµ(α, β, u) = (m,λ, v), T −1µ = T −1µ (m,λ, v) = (α, β, u),Tµ :m =β − αβ + α− 2µ,λ =2µ− α− β2(µ− α)(µ− β),v = (µI − L)u,T −1µ :α = µ− 1λ(1 + m),β = µ− 1λ(1−m),u = (µI − L)−1v,it is possible to write the Fuč́ık spectrum problem Lu = αu+ − βu− as the nonlinearhomogeneous eigenpair problem(µI − L)−1v = λ(v + m|v|). (3.1)Moreover, let us define the corresponding Rayleigh quotient J : L2(Ω) → R asJ(v) =F (v)G(v), where F (v) =12∫Ω(µI − L)−1v · v dx, G(v) = 12∫Ωv2 + m|v|v dx.Due to the homogenity of the problem (3.1), a pair (λ, v) is an eigenpair of (3.1), if andonly if v is a critical point of J and λ = J(v) is the corresponding critical value.Let us consider the Neumann operator LN : dom(LN) ⊂ C([0, π]) → C([0, π]) byLNy := −y′′, dom(LN) :={C2([0, π]) : y′(0) = y′(π) = 0}.Let us set up e.g. µ = 2. Then taking all fixed m ∈ [−1, 1], the minimization andthe maximization processes of functional J(v) generate parts of the second and the firstnontrivial Fuč́ık curves of LN, respectively (see Fig. 3.1). Thus, this approach involveonly those parts of Fuč́ık curves for which (α− µ)(β − µ) > 0. This disadvantage can beovercome if we add e.g. another parameter δ as in the following section.Equadiff-11. The beam operator and the Fuč́ık spectrum 3070 2 4 6 8 10 12 14 160246810121416−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−4−3−2−101234α mβ λµ = 2m = 0.4Fig. 3.1. The first four nontrivial Fuč́ık curves of Σ(LN) transformed by Tµ.4. Revisited variational approach. Let µ ∈ R \ σ(L) and δ ∈ R be such that(µ + δ) 6∈ σ(L). If we take into account the following generalized transformationTµ,δ = Tµ,δ(α, β, u) = (m, λ̃, v), T −1µ,δ = T−1µ,δ (m, λ̃, v) = (α, β, u),Tµ,δ :m =β − αβ + α− 2µ,λ̃ =2µ− α− β2(µ− α)(µ− β)+δ(2µ− α− β),v = (µI − L)u,T −1µ,δ :α = µ− 1−δλ̃λ̃(1 + m),β = µ− 1−δλ̃λ̃(1−m),u = (µI − L)−1v,then the Fuč́ık spectrum problem Lu = αu+ − βu− reads as the nonlinear problem((µ+δ)I − L)−1v = λ̃(v + m(I−δ[(µ + δ)I − L]−1)|v|). (4.1)Due to homogenity of such an eigenpair problem, the critical points of the correspondingRayleigh quotient J : L2(Ω) → RJ(v) =F (v)G(v), F (v) =12∫Ω((µ+δ)I − L)−1v · v dx,G(v) =12∫Ωv2+ m(I−δ[(µ + δ)I − L]−1)|v|v dx,together with their critical values λ = J(v) are in one to one correspondence with eigen-pairs (λ, v) of (4.1).Let us take µ = 1/6 and δ = 4. Then all nontrivial Fuč́ık curves of σ(LN) are locatedin a strip {(m,λ) ∈ [−1, 1]× R} (see Fig. 4.1). As a consequence of this fact, the wholesecond and the whole third nontrivial Fuč́ık curves of Σ(LN) including their asymptotes(for m = ±1) can be obtained by the maximization and the minimization of the functionalJ(v), respectively.308 P. Nečesal0 2 4 6 8 10 12 14 16 18 2002468101214161820−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−6−4−20246α mβ λµ = 1/6δ = 4m = 0.4Fig. 4.1. The first four nontrivial Fuč́ık curves of Σ(LN) transformed by Tµ,δ.5. Example. In order to demonstrate the applicability of our variational approachalso in the case of partial differential operators, let us consider the following Neumannproblem for the wave operator u 7→ −(utt − uxx)utt − uxx + αu+ − βu− = 0, (x, t) ∈ (0, 2)× (0, 1),ux(0, t) = ux(X, t) = 0, t ∈ [0, 1],ut(x, 0) = ut(x, T ) = 0, x ∈ [0, 2].xtuxtuFig. 5.1. The generalized eigenfunction for β = +∞ and the starting eigenfunction for α = β.Let us set up µ = 1 and δ = 0. Starting with the eigenfunction cos πx/2 cos πt, theminimization process of the corresponding functional J(v) for m = 1 leads to λ = 0.8513with ‖J ′(v)‖2 = 2.3·10−5. Then we recover the generalized eigenfunction with α = 1.5873and β = +∞ using the inverse transformation T −1µ,δ (see Fig. 5.1).Equadiff-11. The beam operator and the Fuč́ık spectrum 3096. Conclusion and further research. Our variational approach provides us arobust and global algorithm for exploring the generalized eigenfunctions and particularFuč́ık curves including their asymptotic behaviour. No continuation technique is requiredand presented variational approach is suitable also in the case of the beam operator Lω(see [14] how to effectively implement the inverse operator (µI − Lω)−1). Finally, let usnote that using our variational approach, we explore the qualitatively different behaviourof the Fuč́ık curves and their corresponding generalized eigenfunctions in the case of thebeam operator Lω in contrast with known results concerning the Fuč́ık spectrum forordinary differential operators.7. Acknowledgement. The author acknowledges the support of the Grant Agencyof Czech Republic, project # GA201/03/0671, and the Ministry of Education of the CzechRepublic, Research Plan # MSM4977751301, which made it possible to discuss the topicof this paper in a wider mathematical community.REFERENCES[1] K. Ben-Naoum, C. Fabry, and D. Smets, Structure of the Fuč́ık spectrum and existence of solutionsfor equations with asymmetric nonlinearities, Proc. Roy. Soc. Edinburgh 131 A (2001), 241–265.[2] K. Ben-Naoum, C. Fabry, and D. Smets, Resonance with respect to the Fuč́ık spectrum, Electron.J. Differential Equations, 37 (2000), 1–21.[3] Y. S. Choi, K. S. Jen, and P. J. 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The beam operator and the Fučík spectrum

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The beam operator and the Fučík spectrum

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Institute of Mathematics AS CR, v. v. i.

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