Higher-Order Operators on Networks: Hyperbolic and Parabolic Theory

We study higher-order elliptic operators on one-dimensional ramified structures (networks). We introduce a general variational framework for fourth-order operators that allows us to study features of both hyperbolic and parabolic equations driven by this class of operators. We observe that they extend to the higher-order case and discuss well-posedness and conservation of energy of beam equations, along with regularizing properties of polyharmonic heat kernels. A noteworthy finding is the discovery of a new class of well-posed evolution equations with Wentzell-type boundary conditions.


Introduction
Double-beam systems are a classical subject of theoretical mechanics, see e.g. [9,43]: they consist of two beams mediated by a viscoelastic material layer. On the mathematical level, this is modeled by strong couplings between the equations, usually complemented by identical, homogeneous boundary conditions-say, clamped or hinged. In the last two decades, coupled systems consisting of networks of (almost) one-dimensional beams have aroused more and more interest: unlike in double-beam systems, all interactions take place in the ramification points.
Our aim in this note is twofold: we first discuss some properties of beam equations on networks of one-dimensional elements, with a focus on the solution properties that depend on rather general transmission conditions in the nodes. To this purpose, we propose a variational treatment of beam equations developing a formalism that happens then to be easily extendible to the study of parabolic equations driven by elliptic operators of arbitrary even order, again with general combinations of stationary and dynamic boundary conditions.
The analysis of evolution equations on networks has become a very popular topic since Lumer introduced in [32] a theoretical framework to study heat equations on ramified structures; but in fact, time-dependent Schrödinger equations on networks have been studied by quantum chemists since the 1940s and perhaps earlier, see the references in [41, § 2.5]. Also, networks of thin linear beams have been studied often in the literature, with a special focus on controllability and stabilization issues, ever since the pioneering discussion in [29,30]. The simplest case of a network corresponds to a path graph-a concatenation of linear elements. This case models a beam consisting of different segments with various elasticity properties: different vertex conditions for the bi-Laplacian that appears in the beam equation on a single path graph have been derived from physical principles in [8]. It turns out that there is no unique natural choice for transmission conditions in a network's node: based on physical considerations, several conditions have been proposed in the literature, especially of stationary nature [7,[14][15][16][17]27]. In [24] we have applied the classical extension theory of symmetric positive semidefinite operators to discuss general transmission conditions for the bi-Laplacian; in particular, we have described an infinite class of transmission conditions leading to well-posedness of the beam equation on networks.
Along with stationary conditions, conditions of dynamic type have been very popular in the context of networks of beams, as they naturally model massive junctions: we refer to [35][36][37]42]. In Sect. 2 we are going to continue and extend the analysis of fourth-order operators on finite networks initiated in [24]: we characterize dynamic conditions leading to self-adjoint realizations and in fact, parametrize an infinite class of transmission conditions under which the corresponding system of beams is well-posed and enjoys conservation of energy. Rather general dynamic boundary conditions leading to self-adjoint, dissipative realizations on a single interval have been studied in [20]: they are special cases of our parametrization, too, which elaborates on an idea of Arendt and his co-authors [2,3] for the discussion of second-order elliptic operators with dynamic boundary conditions through quadratic forms on product Hilbert spaces. An interesting feature of our theoretical framework is that, by appropriately choosing the product space, we can simultaneously treat evolution equations endowed by dynamic and/or stationary conditions. We mention that similar ideas and a comparable formalism have been successfully applied in [28] to study hyperbolic systems with dynamic boundary conditions. In Sect. 3 we also extend most of the methods developed for the beam equations to the study of parabolic features of equations driven by jth powers of the Laplacian, again on networks. Linear and semilinear elliptic equations associated with such operators have been discussed often in the literature since a classical article by Davies [12]; we also refer to [5] for a collection of the main features of such equations, to [31] for a deep study of contractivity properties of the generated semigroups, to [23] for a selection of related models in physics and mechanics, and to [13,18,21] for a study of some realizations with dynamic boundary conditions on domains. After briefly showing well-posedness of general hyperbolic equations (second derivative in time, arbitrary integer powers of the Laplacians in space) with dynamic and/or stationary equations we turn to the properties of the analytic semigroup generated by the same differential operator's realizations on a finite network. In particular, we show that such a semigroup is of trace class and (under mild assumptions) ultracontractive; we also show that in spite of failure of the maximum principle for any j ≥ 2, depending on the transmission conditions such a semigroup may or may not be eventually sub-Markovian and eventually enjoy a strong Feller property.
Based on a classical idea that goes back to [19], it is well-known that in the case of second-order elliptic operators there is a direct connection between dynamic and Wentzell-type boundary conditions: formally taking the boundary trace of the evolution equation and plugging it into the dynamic boundary condition, one thus obtains a boundary condition involving boundary terms of order as high as the operator itself. We are also going to demonstrate that the class of Wentzell-type boundary conditions hitherto considered in the literature is unnecessarily restrictive; possibly the most surprising finding of our paper is that the natural Wentzell-type boundary conditions for an evolution equation of order 2j in space are in fact of higher order than the operator itself: namely, of order 3j − 1, see Proposition 3.5.(iv).

The Beam Equation on Networks
We consider a finite connected graph G = (V, E), with V := |V| vertices and E := |E| edges; loops and parallel edges are allowed. We also denote by E v the set of all edges incident in v. We fix an arbitrary orientation of G, so that each edge e ≡ (v, w) can be identified with an interval [0, e ] and its endpoints v, w with 0 and e , respectively. In such a way one naturally turns the G into a metric measure space G: a network (or metric graph) whose underlying discrete graph is precisely G. We refer to [41,Chapt. 3] for details.
Functions on G are vectors (u e ) e∈E , where each u e is defined on the edge e (0, e ). We introduce the Hilbert space of measurable, square integrable Boundary values of elements of L 2 (G) are not defined, and in this sense functions that are merely in L 2 (G) cannot mirror the topology of the network G: in order to describe transition conditions in the vertices we to introduce the Sobolev spaces it is symmetric and strictly positive, hence its self-adjoint extensions can be described by means of the extension theory due to Friedrichs and Krein. An important role is played by the closable quadratic form associated with A, which is given by However, a sesquilinear form can-and typically will-have different associated operators whenever it is studied on different Hilbert spaces. In [24, § 3] we have characterized the self-adjoint extensions of A on L 2 (G) and discussed further realizations that generate cosine operator functions and operator semigroups, again on L 2 (G). In this paper we are going to discuss the more general case of extensions on Hilbert spaces of the form where Y d is any subspace of the "boundary space" C 4E . Therefore, let us consider the space L 2 (G) ⊕ Y d whose elements are of the form u = u θ . This is a Hilbert space with respect to the canonical inner product In the following, we denote by P Z and P ⊥ Z the orthogonal projector of C 4E onto subspaces Z and Z ⊥ , respectively; we also introduce the notation Consider the quadratic form It is closable and its closure is associated with a self-adjoint operator: it is easily seen that this is the operator matrix Consider now A max and A 0 , the maximal and the minimal realizations of the operator A, respectively, endowed with the domains The main result in this section is a characterization of all further self-adjoint extensions of A 0 on This motivates us to introduce the Hilbert space we can hence compactly writẽ (2.4) and all v ∈ V. Summing up, we describe the transmission conditions in the network's vertices by means of a subspace Y of C 4E : this consists of two orthogonal subspaces Y d and Y s that encode the dynamic and stationary part of the transmission conditions, respectively.
Observe that the quadratic formã in (2.3) is symmetric if and only if both D := P Y d RP Y d and S := P Ys RP Ys are self-adjoint.
Let us first focus on the case D = 0. In the case of Y d = {0}, this has been proved in [24]. Theorem 2.1 sharpens the main result in [20] already in the case of an interval (i.e., a graph consisting of a unique edge).

Proof. (i)⇒(ii) Because
A is an extension of the minimal realization A 0 , hence a restriction of the maximal realization A max , A has the same form of the operator matrix in (2.2).
Therefore, the boundary terms should vanish. Considering that Γ from which it follows that S is self-adjoint and v ∈ D(A). Indeed, from the first condition of (2.5) one straightforwardly obtains that Γ • v ∈ Y . From the last one, using the fact that P Ys (Γ  This motivates us to impose the following throughout this section. Let us recall a celebrated result due to J. Kisyński: given a closed, densely defined operator A on a Banach space X, generation of a cosine operator function by A is equivalent to the existence of a space V such that D(A) → V → X and that the part of the operator matrix 0 I A 0 in V ⊕ X generates a strongly continuous semigroup, see [1,Thm. 3.14.11]. In this case, V is unique and is often called Kisyński space in the literature.

Lemma 2.3. Under the Assumptions 2.2, the operator −A associated with the form
Proof. The sesquilinear form a is the same form introduced in [24], whereas V is isomorphic to We have already checked in [24,Thm. 4.3] that a is densely defined and continuous. Let it is clear that this map has dense range, hence a is a j-elliptic form in the sense of [3, § 2], and the associated operator in the sense of [3, Thm. 2.1] agrees with the operator associated with a with domain V. Because We are finally in the position to prove the main result of this section; we re-introduce the boundary term D = P Y d RP Y d which we have been discussing at the beginning of this section, along with further perturbing terms.
Theorem 2.4. Under the Assumptions 2.2 let, for all e ∈ E, p e ∈ L ∞ (0, e ) be real-valued such that p e (x) ≥ P e for some P e > 0 and a.e. x ∈ (0, e ) and let Π be a self-adjoint, positive definite operator on Y d . Then for all D ∈ L(Y d ) with domain Proof. Under our assumptions we can endow L 2 (G) ⊕ Y d with the inner product This is again a Hilbert space and in fact the new inner product is equivalent to the canonical one, hence both Hilbert spaces are isomorphic. Let us first consider the case D = 0. A direct computation similar to that preceding Theorem 2.1 shows thatÃ is the operator associated with a on L 2 (G) ⊕ Y d with respect to the above inner product and we can prove just like in Lemma 2.3 that −Ã is the generator of a cosine operator function on L 2 (G) ⊕ Y d with respect to the canonical inner product, since so it is with respect to the equivalent inner product in (2.7).
In order to complete the proof, it suffices to observe that the sesquilinear form is bounded. Thus, the operator associated with the sesquilinear form a + b in L 2 (G) ⊕ Y d with respect to the inner product in (2.7)-i.e.Ã in (2.6)-is again the generator of a cosine operator function in L 2 (G) ⊕ Y d with respect to the canonical inner product.

Remark 2.5.
(1) We stress that while the case of D = 0 could also be dealt with as a bounded perturbation of a well-behaved operator, the case of S = 0 cannot and requires the specific treatment in Lemma 2.3. (2) One sees thatÃ is self-adjoint-or equivalently the associated sesquilinear formã, i.e., where (C(t, −Ã)) t∈R is the cosine operator function generated by −Ã and (S(t, −Ã)) t∈R denotes the sine operator function generated by −Ã, which is defined by Moreover, our approach based on forms and cosine operator functions allows us to derive the well-posedness of the damped wave equation dx 4 with transmission conditions Γ • u ∈ Y s ; while the latter bi-Laplacian realization is well-behaved by the theory developed in [24], unbounded perturbation theory for cosine operator function is a notoriously tricky business. Theorem 2.4 can also be compared with some results in [20]. For example, all cases where Y d = Y 1 ⊕ {0 C 2E } are covered by Theorem 2.4 but seemingly not by [20,Thm. 8].
Beam equations with dynamic boundary conditions have been often considered in the literature and can be studied with our formalism. In order to make this formalism more concrete we give some examples. (iii) dynamic condition in v 1 : (iv) compatibility condition in v 1 : u (v 1 ) = e∼v1 ∂u e ∂ν (v 1 ).
With the formalism of Theorem 2.1, the above vertex conditions correspond to a second order abstract Cauchy problem (2.9), where the operatorÃ is determined by the choice of the subspaces here 1 Ev denotes the characteristic function of the set E v of edges incident with any given vertex v.
Let us now investigate on conservation of energy for the beam equation. Let us introduce the energy-type functionals in for any solution u = u PY d Γ•u of (2.9), whereã is the sesquilinear form in (2.8).

Lemma 2.8. Under the assumptions of Theorem 2.4, letÃ be self-adjoint and positive semi-definite. Then the total energy E of (2.9) is conserved, i.e., it is a constant (over time) that only depends on the initial data f ∈ D(Ã), g ∈ V.
Proof. Let us first observe that This motivates us to introduce the notation for the total energy of (2.9) with initial data f, g. J.A. Goldstein and coauthors have studied since [25] whether wave-like equations enjoy equipartition of energy, i.e., Proof. Let B be the square root ofÃ; it is well-known that its domain agrees with V. It is known that (2.9) enjoys equipartition of energy if and only if see [25,Thm. and the text around (14)]. Now, because G is finite, V is compactly embedded in L 2 (G) ⊕ Y d and hence B has compact resolvent: accordingly, there exists an orthonormal basis of L 2 (G)⊕Y d of eigenvectors of B. Let λ be an eigenvalue of −B and φ be a corresponding normalized eigenvector: then showing that equipartition of energy fails to hold.
In the proof of Proposition 2.9, the square root B of our operator matrix A has appeared. While the proof relies on general properties of square roots, it is sometimes possible to describe it more closely: this is interesting because it delivers a more explicit formula for the cosine and sine operator functions Example 2.10. 1. In [24] we have reviewed stationary transmission conditions that appear in several models of beam networks in the literatureespecially in [7,8,15,16,27], showing that they fit in our scheme; additionally, we have considered the bi-Laplacian with continuity conditions across the vertices and zero conditions on the first, second, and third derivatives at the endpoints of each edge, and determined its Friedrichs and Krein-von Neumann extensions. All these realizations satisfy the assumptions of Lemma 2.8 and Proposition 2.9, leading to conservation of energy and failure of equipartition of energy. On the other hand, only the transmission conditions in [24,Exa. 3.2], taken from [15], lead to a realization of the forth derivative that is a square operator.

A Laplacian realization on a network with conditions of continuity on
each vertex v 1 , . . . , v n , complemented by dynamic conditions in v 1 and (stationary) Kirchhoff conditions in v 2 , . . . , v n has been studied by the second author and S. Romanelli: we refer to [34] for more details and an overview of earlier appearances of this model in mathematical and biological literature. It can be written as Taking the square of this operator leads to a bi-Laplacian realization that fits the scheme of our Theorem 2.1. Indeed, the square of this Laplacian is precisely the bi-Laplacian presented in Example 2.7.(1).

Parabolic Theory of Polyharmonic Operators with Boundary Conditions on Networks
In this section we are going to extend the theory developed in the previous section to the study of jth powers of the Laplacian, for generic j ≥ 2. It 50 Page 14 of 22 F. Gregorio, D. Mugnolo IEOT turns out that the formalism introduced before allows us to discuss parabolic problems driven by general poly-harmonic operators under very general (stationary or dynamic) boundary conditions. It is easy to prove by induction that for all j ∈ N 0 This clearly suggests to introduce the sesquilinear form for any given subspace Y of C 2j and any linear operator R on Y . This form is symmetric (and hence the corresponding operator is self-adjoint) if and only if R is self-adjoint; indeed, the corresponding operator A is the operator (−1) j d 2j dx 2j with boundary conditions Following the same ideas in the proof of [6, Thm. 1.4.4] one can prove that each self-adjoint realization of (−1) j d 2j dx 2j is of this type. Upon replacing L 2 (0, ) by e∈E L 2 (0, e ) ⊕ Y d , scalar-valued functions by C E -valued functions, and the boundary space C 2j by C 2jE , we can consider the sesquilinear form and then extend the above considerations to the case of elliptic operators of order 2j on networks; the essential ideas coincide with those presented in the previous section and we omit the details.
where the operators Γ • , Γ • are defined in (3.1). Then for any extension A of The operator A takes the form In the non-dynamic case of Y d = {0}, A 0 satisfies zero boundary conditions on all derivatives up to order 2j − 1: hence A 0 is a symmetric, positive definite operator and we recover the classical characterization of self-adjoint extensions of one-dimensional polyharmonic operators.
Motivated by the above result we therefore impose the following in the remainder of this section.
We can thus state the following, without proof. Theorem 3.3. Under the Assumptions 3.2, let Π be a self-adjoint, positive definite operator on Y d . Also, let for all e ∈ E p e ∈ L ∞ (0, e ) be real-valued and such that p e (x) ≥ P e for some P e > 0 and a.e. x ∈ (0, e ). Then for all Remark 3.4. Theorem 3.3 extends the generation results from [40], where only the case of j = 1 and Y s = {0} was considered; in turn, the latter generalized the main assertions from [34], where Y = Y d was taken to be the subspace of C 2E consisting of those vectors that are vertex-wise constant, i.e., v∈V 1 v . , each generator of a cosine operator function also generates on the same Banach space an analytic semigroup of angle π 2 . In the next proposition we study some properties for this semigroup. For the sake of simplicity we focus on the simple case of p ≡ 1 and Π = Id, but see Remark 3.6 below. Proposition 3.5. Under the Assumptions 3.2 the following properties hold for the semigroup generated by (i) e −tÃ is of trace class for all t > 0.
(ii) If D, S are dissipative, then there exist C, ω > 0 such that The terms Γ • u and Γ • u involve differential terms of order up to 2j − 1 and j − 1, respectively. The property in (iv) is therefore surprising: it states that the natural order of the Wentzell-type boundary conditions for an operator of order 2j is not necessarily 2j, as most usually considered in the literature (for example, in [13,18,20,21]) but rather up to 3j − 1 (as it was already observed in [10], for j = 2).
Let u := ( u θ ) ∈ V. The Gagliardo-Nirenberg inequality, cf. [22], yields Therefore, since D and S are dissipative . Observe that S, D dissipative also implies that the semigroup is contractive. Also, recall that if an operator A generates an analytic semigroup, then there exists a positive constant such that ||Ae −tA || ≤ c t for all t > 0. We follow an argument similar to that in [24, Proposition 5.1]: letting u := e −tÃ f and using analyticity and contractivity on L 2 (G) one obtains This concludes the proof.
Observe that from the computations in (3.4) one also finds  [4,44].) The same idea carries over to our setting and yields that the operator matrix In [24] we have discussed the bi-Laplacian on G through extension theory of Hilbert spaces. We could pursue similar results here, but we avoid the details. Suffice it to say that if we impose continuity vertex conditions on the pre-minimal operator, i.e., we consider the operator matrix A in (2.2) restricted to then its Friedrichs extension A F is the realization of A whose domain contains functions u that enjoy the boundary conditions ∂ h ν u = 0 for all 1 ≤ h ≤ j − 1, along with continuity of u on the metric space G and a Kirchhoff-type condition on ∂ 2j−1 ν u at each vertex. In particular, the null space of A F is 1-dimensional: it is given by the space of all constants on G. Also observe that e −tAF maps, for each t > 0, Combining these two facts with [24,Cor. 7.4 and Prop. 7.5] we can immediately deduce remarkable properties of the semigroup generated by −A F , which we state without proof. It is interesting to compare them with the properties of the −A N , defined as the realization of −A whose domain contains functions u such that ∂ h ν u is continuous of G for all 0 ≤ h ≤ j − 1, while a Kirchhoff-type condition is satisfied by ∂ h ν u for all j ≤ h ≤ 2j − 1.  property. On the other hand, the semigroup on L 2 (G) ⊕ Y d generated by −A N is not even individually asymptotically positive.
By eventual sub-Markovian (resp., eventually irreducible) we mean that there exists some t 0 > 0 such that 0 ≤ e −tAF f ≤ 1 (resp., 0 e −tAF f) for all t ≥ t 0 and all f such that 0 ≤ f ≤ 1 (resp, 0 ≤ f, 0 ≡ f), where 1 is the constant 1 function. Also, a bounded semigroup is called individually asymptotically positive if the distance between each orbit and the Hilbert lattice's positive cone tends to 0 as t → ∞.
Similarly, we say that a semigroup eventually enjoys a strong Feller property if for all t ≥ t 0 it is sub-Markovian and maps bounded measurable functions to bounded continuous functions.
While −A F generates on L 2 (G) a semigroup that leaves C(G) invariant and is bounded in ∞-norm, it is currently unknown whether its part in C(G) is the generator of a strongly continuous semigroup.
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