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. [CS95,VOK00]: 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 ∂ 2 u ∂t 2 = −u ′′′′ (here and in the following, ′ = ∂ ∂x ) 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; as a byproduct, we develop a formalism that can be easily extended 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 [Lum80] 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 [Mug14, § 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 [LLS92,LLS93]. The simplest case of a network corresponds to a path graph -a concatenation of linear elements. This case corresponds to 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 [CDKP87]. 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 [DN98,DN99,DN00,BL04,DZ06,KKU15]. In [GM19] 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 dynamical type have been very popular in the context of networks of beams, as they naturally model massive junctions: we refer to [MR08b,MR08a,MR09,TAN13]; networks of strings (i.e., involving the Laplacian instead of the bi-Laplacian) with dynamic boundary conditions have been discussed already in [MR07]. In Section 2 we are going to continue and extend the analysis of fourth-order operators on finite networks initiated in [GM19]: we characterize dynamical 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 [FGGR07]: they are special cases of our parametrization, too, which elaborates on an idea of Arendt and his co-authors [AMPR03,AtE12] 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 study evolution equations endowed by dynamic and/or stationary conditions at the same time.
In Section 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 [Dav95]; we also refer to [GGS10] for a selection of related models in physics and mechanics, and to [FGGR08, FGG + 10] for a study of some realizations with dynamic boundary conditions on domains. After briefly showing wellposedness 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 at least to [FGGR02], it is well-known that in the case of second-order elliptic operators there is a direct connection between dynamic and Wentzelltype 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 3.3.(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, ℓ e ] and its endpoints v, w with 0 and ℓ e , respectively. In such a way one naturally turns the G into a metric measure space G: a metric graph whose underlying discrete graph is precisely G. We refer to [Mug14, Chapt. 3] for details.
Functions on G are vectors (u e ) e∈E , where each u e is defined on the edge e ≃ (0, ℓ e ). We introduce the Hilbert space of measurable, square integrable functions on G L 2 (G) := e∈E L 2 (0, ℓ e ) endowed with the natural inner product 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 metric graph: this motivates us to introduce the Sobolev spaces they consist of L 2 (G)-functions whose k-th weak derivatives are elements of L 2 (G), too. Consider the operator A defined edgewise as the fourth derivative 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 [GM19, § 3] we have characterized the self-adjoint extensions of A on L 2 (G) and discussed further realizations that generate cosine operator function 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 . This is a Hilbert space with respect to the canonical inner product In the following, we denote by P Z the orthogonal projectors of C 4E onto a subspace Z; 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 realisations 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 Hence, consider the sesquilinear form a defined by Integrating by parts we find We deduce for all u ∈ H 4 (G) ∩ V satisfying (2.3) and (2.4) and (2.5) We can hence compactly write Observe that the restriction of a to the space V is symmetric if and only if both R d := P Y d RP Y d and R s := P Ys RP Ys are self-adjoint.
The operator A has the following form In the case of Y d = {0}, this has been proved in [GM19]. Theorem 2.1 sharpens the main result in [FGGR07] even in the case of an interval (i.e., a graph consisting of a unique edge).
Therefore, the boundary terms should vanish. Considering that Γ • u ∈ Y one has Therefore, R d and R s are self-adjoint and v ∈ D(A). Indeed, from the first and second condition one straightforwardly obtains that R d is self-adjoint and that Γ • v ∈ Y , respectively. From the last one, using the fact that P Ys (Γ • u + R s Γ • u) = 0 one obtains that R s is self-adjoint and (ii)⇒(i) In order to prove self-adjointness of A we have to establish two facts: (a) if u and v belong to D(A), then (2.6) holds and (b) if u ∈ D(A) and (2.6) holds, then v ∈ D(A). If u, v belong to D(A), then the set of equalities (2.6) holds and this take cares of (a). If instead u ∈ D(A) and (2.6) holds with R d and R s self-adjoint one directly has, as shown before, that v ∈ D(A) that is (b).
This motivates us to impose the following in this and the following section.
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 We have already checked in [GM19, Thm. 4.3] that H 2 Y (G) 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 [AtE12, § 2], and the associated operator in the sense of [AtE12, Thm. 2.1] agrees with the operator associated with a with domain V. Because where (C(t, −A)) t∈R is the cosine operator function generated by −A and (S(t, −A)) t∈R denotes the sine operator function generated by −A, which is defined by We are finally in the position to prove the main result of this section.
Theorem 2.5. Under the Assumptions 2.2, let for all e ∈ E p e : (0, ℓ e ) → R such that P 1 e ≥ p e (x) ≥ P 0 e for some P 1 e , P 0 e > 0 and all x ∈ (0, ℓ e ). Let Π be a self-adjoint, positive semidefinite operator on Y d , T a linear operator on Y d , and R a linear self-adjoint operator on Y . Then generates on L 2 (G) × Y d a cosine operator function.
Theorem 2.5 can be compared with some results in [FGGR07]. For example, all cases where Y d = Y 1 × {0 C 2E } are covered by Theorem 2.5 but seemingly not by [FGGR07, Thm. 8].
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 T = R d . A direct computation similar to that in (2.5) shows that A 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 inner product in (2.8), hence also with respect to the canonical inner product.
In order to complete the proof, it suffices to observe that Lemma 2.6. Under assumptions 2.2, let a be accretive. Then the total energy E of (2.7) is conserved, i.e., it is a constant (over time) that only depends on the initial data f, g.
Proof. Let us first observe that Differentiating the energy of a given solution u with respect to t and integrating by parts one = 0 for all t ≥ 0 and P Ys (Γ • u + RP Ys Γ • u) = 0. This motivates us to introduce the notation for the total energy of (2.7) with initial data f, g. Goldstein and coauthors have studied since [Gol69] whether wave-like equations enjoy equipartition of energy, i.e., when To this purpose, let us consider a square operator A for R = 0 Indeed, if one considers B as a simple computation yields A = (B) 2 . Now, by [MR06,Corollary 3.14] the operator −B generates a cosine operator function. It is known that C(t, −A) = cosh(t, iB), i.e., showing that equipartition of energy fails to hold. , lead to a realization of the forth derivative that is a square operator.

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 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. We impose the following throughout this section.
It is easy to prove by induction that for all j ∈ N where the vectors Γ • u, Γ • v ∈ C 2j are defined using the notation 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 Indeed, following the same ideas in the proof of [BK13, Thm. 1.4.4] one can prove that each self-adjoint realization of (−1) j d 2j dx 2j is of this type. By replacing L 2 (0, ℓ) by e∈E L 2 (0, ℓ e ), scalarvalued functions by C E -valued functions, and the boundary space C 2j by C 2jE , we can extend these considerations to the case of elliptic operators of order 2j on networks; the essential ideas coincide with those presented in the previous sections and we omit the details. We can thus state without proof the following.
Theorem 3.2. Under the Assumptions 3.1, let Π be a self-adjoint, positive semidefinite operator on Y d , T a linear operator on Y d , and R a linear self-adjoint operator on Y . Also, let for all e ∈ E p e : (0, ℓ e ) → R such that P 1 e ≥ p e (x) ≥ P 0 e for some P 1 e , P 0 e > 0 and all x ∈ (0, ℓ e ). Then generates on L 2 (G) × Y d a cosine operator function.
Each generator of a cosine operator function on a Hilbert space H also generates on H 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 Π = 1 and T = P Y d RP Y d , but see Remark 3.4 below. Proposition 3.3. Under the Assumptions 3.1, let R be a linear self-adjoint operator on Y . Then the following properties hold for the semigroup generated by (i) e −tA is of trace class for all t > 0.
(ii) If R d , R s are dissipative, then there exist C, ω > 0 such that e ωt for all t > 0.
(iii) e −tA has for all t > 0 an integral kernel of class L ∞ .
x ∈ G, satisfies the boundary condition 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 usually considered in the literature (see e.g. [FGGR07]), but rather up to 3j − 1.
Proof. (i) Observe that the image of the form domain V under j is compactly embedded in L 2 (G) × Y d , hence the operator −A has compact resolvent. Indeed, more is true: by [Gra68] the embedding of j(V) in L 2 (G) × Y d is of Schatten class, hence the analytic semigroup generated by −A consists for all t > 0 of trace class operators [MN12,Rem. 3.4].
(ii) Let us then consider L ∞ (G) × Y d with the norm Let u := ( u θ ) ∈ V. The Gagliardo-Nirenberg inequality, cf. [Gag59], yields and recursively Therefore, since R is dissipative Observe that R dissipative also implies that the semigroup is contractive. We follow an argument similar to that in [GM19, Proposition 5.1]: letting u := e −tA f and using analyticity and contractivity on L 2 (G) one obtains This concludes the proof.
(iii) In particular, the same proof as in (ii) shows that the adjoint semigroup satisfies the same ultracontractivity estimate; hence by duality for all t > 0, and in particular e −tA maps for all t > 0 L 1 to L ∞ : the existence of an L ∞ -kernel then follows from the Kantorovich-Vulikh Theorem.
(iv) Because of the smoothing effect of the analytic semigroup generated by −A, the solution u(t, ·) is for all t > 0 infinitely often differentiable (with respect to space), hence we can take the boundary values Γ • d 2j dx 2j u of d 2j dx 2j u. Because the time derivative and Γ • commute, plugging the parabolic equation satisfied in the interior of the edges into the dynamic boundary conditions we deduce that Observe that from the computations in (3.1) one also finds Remark 3.4. Semigroups generated by Laplacians on networks with dynamic vertex conditions have been studied in [MR07]. It has been observed in [MR07, Rem 3.6] that modifying the coefficients of the normal derivative (the lower-left entry of the relevant operator matrix in that context) amounts to a relatively compact perturbation of an analytic semigroup generator: by a perturbation theorem due to Desch and Schappacher, the new operator generates an analytic semigroup, too. (Similar assertions were proved in [BBR06,VV08].) The same idea carries over to our setting and yields that the operator matrix In [GM19] 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 C ∞ c (0, ℓ e ) and θ = Γ • u , 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 −tA F maps, for each t > 0, L 2 (G) × Y d into D(A F ) ֒→ u θ ∈ C(G) × Y d : u ∈ e∈E C 2j−1 ([0, ℓ e ]) and θ = Γ • u .
Combining these two facts with [GM19, Cor. 7.4 and Prop. 7.5] we can deduce interesting properties of the semigroup generated by −A F . It is interesting to compare them with the properties of the realization −A N , defined as the realization 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.
Proposition 3.5. Let j ≥ 2. Under the Assumptions 3.1, for all subspaces Y d of Y the semigroup on L 2 (G)×Y d generated by −A F is uniformly eventually sub-Markovian; furthermore, it eventually enjoys a uniform strong Feller property. On the other hand, the semigroup on L 2 (G)×Y d generated by −A N is not even individually asymptotically positive positive.
By eventual sub-Markovian (resp., eventually irreducible) we mean that there exists some t 0 > 0 such that 0 ≤ e −tA F f ≤ 1 (resp., 0 ≪ e −tA F 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 an strong Feller property if for all t ≥ t 0 it is sub-Markovian and maps bounded measurable functions to bounded continuous functions.
Currently we do not know whether the part of −A F in the closure of its domain, which is (isomorphic to) C(G), generates on C(G) a strongly continuous semigroup.