Finite Mechanical Proxies for a Class of Reducible Continuum Systems

We present the exact finite reduction of a class of nonlinearly perturbed wave equations, based on the Amann-Conley-Zehnder paradigm. By solving an inverse eigenvalue problem, we establish an equivalence between the spectral finite description derived from A-C-Z and a discrete mechanical model, a well definite finite spring-mass system. By doing so, we decrypt the abstract information encoded in the finite reduction and obtain a physically sound proxy for the continuous problem.


Introduction
In this paper we revisit some aspects of a rather classical analytical model occurring in many physical settings, the following stationary nonlinear problem: where Ω is an open domain in R n with piecewise smooth boundary, u ∈ H ∶= H 1 0 (Ω; R), −Lu is elliptic and F (u) is a nonlinear operator, with ε > 0 a perturbation parameter. Here −λu represents a sort of linear soft device, possibly breaking, for λ > 0 large enough, the elliptic character of −Lu − λu. Stationary elastic membranes, reaction-diffusion, Ginzburg-Landau, among others, can be settled into this format.
First at all, we recall that if N ′ (u) is symmetric with respect to the L 2 scalar product, (F ′ (u) actually) Volterra-Vainberg theorem applies (see [23], [4]), and we can write the energy functional: and obtain a variational formulation for (1): dE(u) = 0. Then, we apply to E(u) a version of Amann-Zehnder (AZ) exact finite reduction, which is a global Lyapunov-Schmidt technique, introduced in Hamiltonian dynamics in [2] and [10], and in fields theory in [1] (see also [5,6,18]), (1). In such a way, we reduce the original variational problem to a finite dimensional system of 'algebraic' equations: the energy functional (2) is shrunk to a m variables functionẼ ∶ R m → R, whose critical points, i.e., dẼ(x) = 0, correspond exactly to the solutions of (1). We will introduce some appropriate conditions on F in order to prove that the reduced functionalẼ(x) is a generating function quadratic at infinity (GFQI), i.e. it coincides, or corresponds in a suitable sense (see details in Section 4), with a non-degenerate quadratic form x ⊺ Qx outside a compact set. 1 This is crucial for our aim: indeed, proving that the sublevel sets diverse topological techniques 2 have been applied. Even though the above abstract settings allow for very general results of existence and multiplicity, here we propose a fairly self contained introduction to AZ and we adopt some stronger hypotheses on F , transforming the variational functional (2) to a finite GFQI, which finally is well arranged towards numerical applications, as e.g. already implemented in [6].
In the last part of the present work (Section 6), we point out the interplay among discrete/continuous and approximate/exact aspects in problem (1), as synthetically illustrated in the following diagram: As a starting point, we recall that, for instance, the elastic string can be modeled by means of a chain of N beads connected by springs of length h and sending N → ∞ and h → 0 keeping mass and strength densities constant (see [13,Chapter 4.]). Here, the continuous system modeled by (1) can be thought as the thermodynamic limit of a large collection of interconnected small particles. Then, by applying AZ, we get back to a finite dimensional problem where approximations are absent, in the sense that the full nonlinear complexity of the set of equilibria is unaltered. However, as described in Section 2, the reduced system is written in spectral variables x, therefore most of the mechanical sense of the starting problem is encrypted in an abstract representation.
In order to tailor a physically meaningful interpretation toẼ(x), we consider its leading term, the quadratic part. Surely, it is possible to find a plenty of finite linear physical systems whose potential energy corresponds precisely to this quadratic part. In Section 6.2, we suggest a suitable linear coordinate 3 transfor-mation x ↦ y = Υ x, selecting, among many variationally equivalent systems, the one mimicking most faithfully the linear part of the original mechanical problem. More precisely, by choosing the above variables y, we pick out the fittest elastic chain from the thermodynamic limit sequence converging to the linear part of (1).
By doing so we provide a further motivation for the AZ reduction coming from the microphysics of the matter. This excursion gives in a sense an accomplishment to a line of thought started in [19].
2 The Amann-Zehnder finite reduction for fields theory Let L be an elliptic operator, i.e., The scalar product on H, defined as is positive definite, indeed: As a result, we will take u 2 ∶= ⟪u, u⟫ as the norm on H.
The eigenfunctions associated to the eigenvalues of the elliptic operator L form a basis for the space H, −Lû j = λ jûj , We are able now to define the Green operator of L, g ∶ H → H, g = (−L) −1 , It is clear that For every fixed m ∈ N, we can consider the following decomposition of H: We will briefly write and we will refer to µ ∈ P m H as to the finite head of v, whereas to η ∈ Q m H as to the infinite tail of v.
If we substitute the relation u = gv in the Dirichlet problem (1): which solutions are the fixed points of the map: The map (I − λg) −1 is well defined for λ ≠ λ j , j = 0, 1, . . . and can be written as The cut-off decomposition splits the problem into a finite and an infinite part: These equations can also be equivalently rewritten as follows, Under appropriate assumptions on F , (10) is uniquely solved for η. Indeed, we can prove that Proposition 1 Let F ∶ H → H be Lipschitz, then for every fixed ε > 0 and µ ∈ P m H there exists m ∈ N such that is a contraction.
Proof First, we notice that η ↦ g(µ + η) is a Lipschitz map with constant 1 λ m+1 . We deduce that (I − λg) −1 is bounded, indeed and clearly sup j By assumption, F is Lipschitz with constant C 2 , therefore also the map (11) is Lipschitz: For sufficiently large m, we conclude that εC 1 C 2 λ m+1 < 1.
If we assume the Gateaux derivative N ′ (u) of the non linear operator considered in (1) to be symmetric with respect to the L 2 -scalar product, i.e.
the Volterra-Vainberg theorem allows us to write an energy functional, and the corresponding variational principle dE(u) = 0 is equivalent to the original Dirichlet problem (1). More precisely, Theorem 1 Every critical point of E is a solution of (1), and viceversa, i.e. Proof The finite parameters reduction of the preceeding section can be applied to the functional E(u) to obtain a finite parameters variational principle, which is equivalent to the infinite dimensional one, i.e. every critical point of E is also a critical point forẼ This equivalence between the finite parameters variational principle d dµẼ (µ) = 0 and the original Dirichlet problem (1) is resumed and proved in the following Theorem 2 There is a one-to-one correspondence between the critical pointsμ ofẼ(µ) and the solutions u of the Dirichlet problem (1).
Proof Plainly, (10) u=g(µ+η(µ)) , g(dµ) We used the facts that (i) the diagonal isomorphism g maps P m H and Q m H into themselves and that (ii) P m H and Q m H are orthogonal also with respect to the L 2 scalar product: We can conclude that 4 Quasi-quadratic functions and their properties

The deformation Lemma
Standard cohomology on a differentiable manifold M , is generalized to the relative cohomology by where N is a submanifold (possibly with boundary) of M and ι ∶ N ↪ M is the inclusion map.
It is well known that the compactness hypothesis for the classical calculus of variations can be weakened by adopting the Palais-Smale condition on the pair (M, f ): Under this condition the critical levels are still compact, even if it could not be the case for M . It is possible to associate a critical value γ(α, f ) to every non zero cohomology class α, i.e., is a critical value for f .
Proof If c is not a critical value, by Palais-Smale condition, there exists ε > 0 such that f −1 ([c − ε, c + ε]) does not contain critical points. It is possible to deform f c+ε in f c−ε , by means of the flow of the vector field which could be smoothly extended to all M to the zero vector field outside of a neighborhood of Note that different cohomology classes could correspond to the same critical value.

Quasi-quadratic functions
The theory of the Generating Functions Quadratic at Infinity (GF QI) has been developed by M. Chaperon, J.C. Sikorav and C. Viterbo, in the context of Hamiltonian dynamics (see [7,8,15,20,21,22,24]). In the symplectic topology environment, D. Théret [22], and later C. Golé [15], revised carefully results claimed by C. Viterbo [25] adopting apparently more general definitions for GFQI. In the following we will stick to the Viterbo-Théret format.
Definition 2 (GFQI) We say that a function S ∶ R n → R is quadratic at infinity (GFQI) if there exists a non-degenerate quadratic form ⟨Qu, u⟩ and a constant K > 0 such that For this class of functions the Palais-Smale property does hold, and for sufficiently large c > 0, the sub-level sets S c ∶= {x ∶ S(x) ≤ c} of a GFQI function S are diffeomorphic to the sub-level sets of the related quadratic form Q, leading as a consequence to the equivalence of the relative cohomology classes: .
In this Section we will give an explicit proof of the equivalence S ±c ≅ Q ±c along a rather different line of thought than [22]. We start by noticing that all the critical points of S are in a compact neighborhood of the origin. Indeed, Moreover, GF QI are Palais-Smale. Indeed, let us consider a P-S sequence and by quasi-quadraticity, 2Qx i < 2K, then x i ∈ B 0, K min Spec Q . Being the sequence definitively confined in a compact set, it is possible to extract a converging subsequence. We are ready now for proving the main result of this section.
Theorem 4 For c > 0 sufficiently large, Proof Assume that c is large enough to surpass every critical value of S, and that at least one of the eigenvalues of Q is positive, otherwise we easily get S c = Q c = R n . Because of (17), we have

and also
Therefore we only need to prove and setting By doing so the derivative of f : is non vanishing if ϕ ′ is bounded, i.e., ϕ ′ < . A continuous ϕ can be defined as follows: (see Figure 2 wherever ϕ is differentiable. An everywhere smooth Urysohn function can be obtained from ϕ by mollification. In the Appendix we provide an explicit construction and verify that the constraint on the derivative still holds. It remains to prove that Q ′ restricted to {Q = c} is > K, provided that c is sufficiently large. We set v ∶= Q ′ (x) = 2Qx and the problem is to minimize v 2 2 for v ⊺ Q −⊺ 4 v = c. Lagrange multipliers format gives: i.e., Therefore, v = 2Qx must be searched among the eigenvectors of Q −⊺ , or equivalently x among the eigenvectors u i of Q. If x = αu i on the level c, i.e., α = c λ i , then we have Since (see, e.g., [14], page 188), we have, for c sufficiently large, where also we have used the fact that (relative) cohomology is invariant under diffeomorphisms. Thus, there exists a critical value for S between −c and c, corresponding -by Theorem 3-to the generator of H ⋆ (Q c , Q −c ).

Quasi-quadratic reduced energy functional
In this section we will investigate around suitable conditions on F in (1), in order that the associated reduced variational principle being a generating function quasi quadratic at infinity, thus admitting at least a critical point. We write more precisely (13) pinpointing the quadratic and the nonquadratic part in u: The reduced functional (15) splits consequently, and, furthermore, a quadratic form in the µ j 's can be isolated: In the general case, there is no reason forẼ L+λ nq (µ) andẼ F (µ) to be quadratic. With an additional hypothesis on F , it is possible to prove the following theorem.
Theorem 5 Let the nonlinear functional F ∶ H → H be a Nemitski operator, F (u) = f ○ u, associated to f ∶ R → R. If f , its derivative f ′ and one of its primitivesf are bounded, then, the reduced functionalẼ(µ) is a GFQI, i.e.
As a result, problem (1) admits at least one solution.

Remark 1
The request on f to be bounded along with its derivative and primitive could seem at first overly restrictive to include sufficiently general nonlinearities. However, it should be noticed that these restrictions apply to the strictly nonlinear part of the operator, and we point out that our class of nonlinearities include C ∞ 0 functions, Schwartz functions, as well zero-mean periodic functions, e.g., sin(x).
Proof Because of (20) and (21), we have that therefore we have to verify thatẼ L+λ nq (µ) andẼ F (µ) are bounded along with their derivatives. Let K > 0 be the constant such that Being λ fixed, while m can be chosen arbitrarily large, it is non restrictive to suppose λ ≪ λ j , ∀j > m.
To give an estimate on the first derivatives ofẼ F and E L+λ nq , we start by considering η ′ (µ) and u ′ (µ). Because of the fixed point equation definingη, we have that where the rightmost fraction is well defined and positive for m sufficiently large. We obtain also an estimate for u ′ , it is easy to check that

A model for the nonlinear elastic string
In this section we consider a specific continuous problem, whose linear part could be obtained as the thermodynamic limit of a sequence of discrete mechanical systems. After having applied AZ reduction, by means of a suitable change of variables, we restore a discrete mechanical system (i) variationally equivalent to the continuum and (ii) physically comparable to an element of the sequence involved in the above thermodynamic limit.
Let us consider the one dimensional stationary problem of the nonlinear elastic string: where u is the vertical displacement of an infinitesimal element of the string, F is, as before, a globally Lipschitz nonlinear Nemitski operator, σ ≥ 0 or < 0 represents a vertical attractive or repelling elastic force, while τ > 0 controls the elastic attraction among neighboring infinitesimal pieces of the string. The eigensystem of −τ ∂ 2 ∂x 22 is given by The above AZ construction leads to the finite variational principle dE(µ) = 0, where This exact finite spectral formulation does correspond to the description of a large number of more or less realistic finite dynamical systems. The direct reconnaissance of (26) shows a system of N harmonic oscillators, whose positions are described by the spectral variables µ j , with elastic constants λ j +σ λ j ; the term E nlin (µ) represents a further nonlinear coupling, incorporating all the nonlinearities of the system.
Our aim is to look for a mechanical discrete system, described by genuine physical variables, such that, at least when the linear part is concerned, its diagonalization -or spectral representation-is proposing precisely the quadratic part E lin (µ) of (26). Among the large number of possible systems satisfying this requirement, we will pick the fitting element from the thermodynamic sequence converging to the linear part of the continuous system (24).
According to this program, we consider the elastic chain of N + 2 oscillating beads (see [13,Chapter 4] and [16]) with fixed extrema: where y i represents the i-th vertical displacement, and the finite differences Laplace operator : represents the bond among neighboring beads. Each oscillator is constrained to move only vertically, is bound to the neighboring beads with a linear spring with elastic constant τ and also to its rest position at y i = 0 with another spring with constant σ. In analogy with our continuous problem, we also introduce a nonlinear Lipschitz force F depending on the vertical displacements y.
The role of the variational principle is played by the potential energy, which can be written as Next, we want to exhibit a suitable linear change of variables such that the potential energy (29) assumes the form of (26). More precisely, we will choose Υ in order to make coincide the quadratic parts of (29) and (26): and the remainder representing the overall nonlinear coupling will be defined consequently: Doing so, we would have obtained a discrete system variationally equivalent to the nonlinear continuous model and with additionally a strong physical correspondence, in the sense that the discrete -described by (29), and in (39) belowsimulates the microscopic granular structure of the continuum.
We carry over explicit computations in the following.

Equivalence among a chain of springs and the elastic string
We can compute the normal modes of the linear part of the system by providing the full solution set of the homogeneous version of equation (27), which is obtained via the eigensystem of the finite difference Laplace operator (28): It is possible to write the potential energy (29) in spectral coordinates ζ k , obtaining an expression analogous to (26): where nowÃ is diagonal. Approaching the thermodynamic limit (for N large), the ratio k 2(N +1) ≃ 0 for any finite k. Therefore we can substitute the sinus with its argument in (33): Therefore, for large N , the first eigenvalues of the chain approximate the first eigenvalues of the continuous elastic string (25).
If we further apply the following diagonal change of coordinates: we obtain a potential energy where the quadratic part now coincides with E lin (µ) in (26). For the good definition of this transformation σ has to be chosen such that the ratio σ+λ k σ+λ k > 0 for every k. In other words, if σ is negative and λ i < −σ < λ i+1 , −σ should be placed among the corresponding pair of eigenvalues for the chain: λ i < −σ < λ i+1 . If σ is positive it suffices that also σ be positive.
Finally, composing the two linear coordinate changes (34) and (36): and by substituting in (26), we obtain the finite dimensional non nonlinear energy function coinciding with the potential energy of a chain of N + 2 nonlinearly coupled oscillators with fixed extrema as in (29): U F (y) ∶= E nlin (µ(y)) = E nlin (Υ −1 y). (39)

Appendix: A smooth Urysohn function
We provide an explicit construction of an everywhere differentiable version of the Urysohn function in (18). Proof By composing Q(x) with a suitable differentiable function g ∶ R → [0, 1], strictly increasing in the interval [c, c + M ], we will maintain the same sublevel sets, and furthermore we are able to control the magnitude of the derivatives. In order to define g, we divide the interval [c, c + M ] in three subintervals, as in Figure 3: if c + M ≤ t. g 2 (and similarly g 4 ) should fill the gap with continuous derivatives: