Linear Hyperbolic Systems on Networks

We study hyperbolic systems of one-dimensional partial differential equations under general, possibly non-local boundary conditions. A large class of evolution equations, either on individual 1-dimensional intervals or on general networks, can be reformulated in our rather flexible formalism, which generalizes the classical technique of first-order reduction. We study forward and backward well-posedness; furthermore, we provide necessary and sufficient conditions on both the boundary conditions and the coefficients arising in the first-order reduction for a given subset of the relevant ambient space to be invariant under the flow that governs the system. Several examples are studied.


Introduction
This paper is devoted to the study of systems of partial differential equations in 1-dimensional setting, more precisely, on collections of intervals: not only internal couplings are allowed, but also interactions at the endpoints of the intervals. It is then natural to interpret these systems as networks, and in fact, we will dwell on this viewpoint throughout the paper.
Partially motivated by investigations in quantum chemistry since the 1950s, differential operators of second order on networks have been often considered in the mathematical literature since the pioneering investigations by Lumer [44] and Faddeev and Pavlov [53]: in these early examples, either heat or Schrödinger equations were of interest. This has paved the way to a manifold of investigations, see e.g. the historical overview in [48].
The equations we are going to study in this paper will, however, be rather hyperbolic; more precisely, the hyperbolic systems of partial differential equations of our interest are of the forṁ where, here and below, we denote everywhere byu and u ′ the partial derivative of a function u with respect to the time variable t and to the space variable x, respectively.
Each of these equations models a physical system: we consider several of these systems and allow them to interact at their boundaries, thus producing a collection of hyperbolic systems on a network. Hyperbolic evolution equations of different kinds taking place on the edges of a network have been frequently considered in the literature, we refer to [49,39,18,20,48] for an overview. Let us emphasize that we shall only consider linear systems: for a survey on some recent developments of the theory for nonlinear hyperbolic systems and many practical applications see e.g. [11].
On each edge of the network we allow for possibly different dynamics (say, Dirac-like, wave-like, beam-like, etc.), thus it would be more precise to write where E is the edge set of the considered network. In particular, in the easiest cases M e may be a diagonal matrix of coefficients of a transport-like equations, but M e may well have off-diagonal entries, or even have a symplectic structure: additionally, we allow all these M e 's to have different size, which of course has to be taken into account by the boundary conditions. We are not going to assume the matrices M e to be either positive or negative semidefinite -in fact, not even Hermitian; therefore, it is at a first glance not clear at which endpoints the boundary conditions should be imposed at all. Indeed, the choice of appropriate transmission conditions in the vertices of the network is the biggest difficulty one has to overcome.
While Ali Mehmeti began the study of wave equations on networks already in [1], it was to the best of our knowledge only at the end of the 1990s that first order differential operators on networks began to be studied. In [14], Carlson defined on a network the momentum operator -i.e., the operator defined edgewise as ı d dx -and gave a sufficient condition -in terms of the boundary conditions satisfied by functions in its domain -for self-adjointness, hence for generation of a unitary group governing a system of equations (1.2)u = ±u ′ with couplings in the boundary (i.e., in the nodes of the networks). Similar ideas were revived in [21,25], where different sufficient conditions of combinatorial or algebraic nature were proposed. A characteristic equation and the long-time behavior of the semigroup governing (1.1) as well as further spectral and extension theoretical properties were discussed in [32,31], respectively, in dependence of the boundary conditions. While all the above mentioned authors -as well as the present manuscript -apply Hilbert space techniques, a semigroup approach to study simple transport equations in Banach spaces (like the space of L 1 -functions along the edges of a network) was presented in [36,46,20], see also [6,Sec. 18] and the references given there.
All these above mentioned papers treat essentially the same parametrization of boundary conditions, namely for a suitable matrix T (possibly consisting of diagonal blocks that correspond to the network's vertices), where u(0) and u(ℓ) denote the vectors of boundary values of u at the initial and terminal endpoints of all intervals, respectively. Bolte and Harrison studied in [8] the Dirac equation on networks. The 1D Dirac equation consists of a system of two coupled first order (both in time and space) equations, much like (1.1); the matrix M e is Hermitian, which allows for simple integration by parts and, in turn, for the emergence of a convenient symplectic structure. Both internal and boundary couplings had to be considered, and the relevant coupling matrix is indefinite. They thus adopted the parametrization Au(0) + Bu(ℓ) = 0 for the boundary conditions, for suitable matrices A, B: mimicking ideas from [34], they were able to characterize those A, B that lead to self-adjoint extensions. Self-adjointness of more general first-order differential operator matrices has been studied in [58].
In this paper, we opt for yet another parametrization of the boundary conditions, inspired by a classical Sturm-Liouville formalism borrowed by Kuchment to discuss self-adjoint extensions of Laplacians on networks in [37] (see also [50] for the "telegrapher's equation" on networks with similar boundary conditions). More precisely, we impose boundary conditions of the form for a subspace Y of the space of boundary values; and find sufficient conditions on Y that, in dependence on M and an auxiliary matrix Q, guarantee that the abstract Cauchy problem associated with (1.1) is governed by a (possibly unitary, under stronger assumptions) group, or a (possibly contractive, under stronger assumptions) semigroup. The auxiliary matrix Q -often called a Friedrichs symmetrizer in the literature, see [7, Def. 2.1] -will play a fundamental role in our approach. Roughly speaking, its role is not to diagonalize M, but only to make it Hermitian; this is done by suitably modifying the inner product of the L 2 -space over the network by means of Q, which therefore has in turn to be positive definite; especially for this reason, our whole theory is essentially relying upon the Hilbert space structure. Our approach allows us in particular to prove generation of unitary C 0 -groups and contractive C 0 -semigroups (and, by perturbation, of general C 0 -(semi)groups). This has a long tradition that goes back to Lax and Phillips [41], who already propose the idea of transforming boundary conditions into the requirement that at each boundary point v the boundary values belong to a given subspace Y v . Indeed, while our well-posedness results are not surprising once the correct boundary conditions are found, the actually tricky task -as long as M is not diagonalizable, the standard assumption among others in [7,30,57,22,5] -is to actually find the right dimension of the space Y v . In this paper, we pursue this task by a fair amount of linear algebra that eventually allows us to parametrize the boundary conditions leading to contractive (semi)groups. This should be compared with the more involved situation in higher dimension, see e.g. [56], which allows for less explicit representation of the boundary conditions. Our setting is thus arguably more general than the approaches to hyperbolic systems on networks that have recently emerged, including port-Hamiltonian systems [62,29,60] and hyperbolic systems that can be transformed into characteristic forms via Riemann coordinates [5], both based on diagonalization arguments.
A relevant by-product of our approach is the possibility to characterize in terms of Q, M, N positivity and further qualitative properties of the solutions of the initial value problem associated with (1.1). In this context, we regard as particularly relevant Proposition 4.9 and Lemma 4.11, which roughly speaking state that the semigroup governing (1.1) can only be positive if M is diagonal, up to technical assumptions (including that Q is diagonal too; this is not quite restrictive, as e.g. all of the examples we will discuss in Section 5 will satisfy it); this negative result essentially prevents most evolution equations of non-transport type arising in applications from being governed by a positive semigroup.
Let us sketch the structure of our paper. In Section 2 we present our general assumptions and discuss their role by showing that a broad class of examples fits into our scheme. In Sections 3 and 4 we then show that our description of boundary condition allows for easy description of realizations that generate (semi)groups. We also find necessary and sufficient conditions for qualitative properties of these semigroups, including reality and positivity.
We conclude this paper by reviewing in Section 5 several applications of our method; among other we discuss forward and backward well-posedness of different equations modeling wave phenomena on networks, including 1D Saint-Venant, Maxwell, and Dirac equations. We study different regimes for the Saint-Venant equation and discuss transmission conditions in the vertices that imply forward, but not backward well-posedness of the Dirac equation. We also study in detail an interesting model of mathematical physics for heat propagation in supercold molecules; we extend the results from [55] by providing physically meaningful classes of transmission conditions implying well-posedness and proving nonpositivity of the semigroup governing this system. Some technical results, which seem to be folklore, are recalled in the appendices.
Acknowledgment. The authors would like to thank Roland Schnaubelt (Karlsruhe) for interesting suggestions concerning early literature devoted to the topic of hyperbolic systems.

General setting and main examples
Let E be a nonempty finite set, which we will identify with the edges of a network upon associating a length ℓ e with each e ∈ E. To fix the ideas, take e ∈ E and ℓ e > 0. (We restrict for simplicity to the case of a network consisting of edges of finite length only, although our results can be easily extended to the case of networks consisting of finitely many leads -semi-bounded intervals -attached to a "core" of finitely many edges of finite length.) We will consider evolution equations of the form where u e is a vector-valued function of size k e ∈ N 1 := {1, 2, . . .}, and M e and N e are matrix-valued functions of size k e × k e . We will couple equations (2.1) for different e ∈ E via boundary conditions given later on.
If M e (x) is Hermitian for all x, then integrating by parts we obtain for all which -provided M ′ e is essentially bounded -allows for an elementary dissipativity analysis of the operator that governs the abstract Cauchy problem associated with (2.1) in a natural Hilbert space. Also the case of diagonalizable matrices M e is benign enough, see e.g. [24, § 7.3]. In the case of general M e , however, it is not easy to control all terms that arise when integrating against test functions and we have to resort to different ideas.
Assumptions 2.1. For each e ∈ E the following holds.
(1) The matrix M e (x) is invertible for each x ∈ [0, ℓ e ] and the mapping [0, and Q e (x)M e (x) are Hermitian for all x ∈ [0, ℓ e ], (ii) Q e (·) is uniformly positive definite, i.e., there exists q > 0 such that If M e (x) is Hermitian for all x, then Assumptions 2.1.(3i) is trivially satisfied by taking Q e to be the k e × k e identity matrix, although this is not the only possible choice and, in fact, it is sometimes actually possible and convenient to take non-diagonal Q e . Assumptions 2.1.3 holds if and only if the system (2.1) is hyperbolic in the sense of [5], see Lemma A.1 below. But we prefer this formulation because the matrices Q e will be involved in the boundary conditions. The fact that Q e (x)M e (x) is Hermitian for all x greatly simplifies our analysis. At the same time, many examples from physics, chemistry, biology, etc., fit in this framework.
The most trivial examples are obtained by taking M e as a diagonal matrix with spatially constant entries: this choice leads to classical (vector-valued) transport problems on networks. For k e ≡ 1 they were considered in [36] and subsequent papers, cf. the literature quoted in [6,Sec. 18].
This system also models arterial blood flow [10,16] for which p is the pressure and q the flow rate at (x, t), L = 1 where A > 0 is the vessel cross-sectional area, C > 0 is the vessel compliance, ν ≥ 0 is the kinematic viscosity coefficient (ν ≈ 3.210 −6 m 2 /s for blood) and α > 1 is the Coriolis coefficient or correction coefficient (α = 4/3 for Newtonian fluids, while α = 1.1 for non-Newtonian fluids, like blood).
Given L, P ∈ C, the Assumptions 2.
where h is the water depth and u the water velocity, and corresponds to the linearization around a steady state (H, V ) of the Saint-Venant model, that in particular satisfies Finally N e is clearly given by (2.9) .

Parametrization of the realizations: the isometric case
The catchiest application of our general theory arises whenever we discuss hyperbolic equations (or even systems thereof) on networks (also known as metric graphs in the literature); in this case, it is natural to interpret E as a set of intervals, each with length ℓ e ; and boundary conditions in the endpoints 0, ℓ e turn into transmission conditions in the ramification nodes. Indeed, for each edge e, k e boundary conditions are required. In general, they are expressed in Riemann (characteristic) coordinates, see for instance [5]. This means that system (2.1) is transformed into a diagonal system with k + e (resp. k − e ) positive (resp. negative) eigenvalues with k + e + k − e = k e and k + e (resp. k − e ) boundary conditions are imposed at 0 (resp. ℓ e ), which allows to fix the incoming information. Here, we prefer to write them in the original unknowns. Furthermore it is a priori not clear how these conditions should be adapted to the case of a network, so we will conversely try to parametrize all those transmission conditions in the network's vertices that lead to an evolution governed by a semigroup (of isometries).
In particular, we are going to look for dissipativity, hence m-dissipativity of the operator ±A whose restriction to the edge e is given by (3.1) (Au) e := M e u ′ e + N e u e in a natural Hilbert space, see (2.1).
To begin with, let us impose the following assumption.
Assumption 3.1. Let G = (V, E) be a finite network (or metric graph) with underlying (discrete) graph G, i.e., G = (V, E) is a finite, directed graph with node set V and edge set E and each e ∈ E is identified with an interval (0, ℓ e ) whereby the parametrization of the interval agrees with the orientation of the edge; the set of these intervals is denoted by E.
We are going to study the problem (2.1) in the vector space Clearly, L 2 (G) becomes a Hilbert space once equipped with the inner product (where · denotes the inner product in C ke ), which is equivalent to the canonical one. Of course, if M e is diagonal, then Assumptions 2.1.(3i) are satisfied e.g. whenever Q e (x) is the identity for all x; however, Example 2.2 shows that Q e M e may be Hermitian even when M e is not. It thus turns out that such an alternative inner product is tailor-made for the class of hyperbolic systems we are considering. The main reason for restricting to the Hilbert space setting is that checking dissipativity in L p -spaces is less immediate.
In order to tackle the problem of determining the correct transmission conditions on A, let us first introduce the maximal domain We want to explicitly state the following, whose easy proof we leave to the reader. Recall that invertibility of M e (x) is assumed for all x ∈ [0, ℓ e ].

Lemma 3.2. It holds
and therefore D max is densely and compactly embedded in L 2 (G).
We stress that compactness of the embedding can actually fail if 0 is an eigenvalue of M e (·) at the endpoints of (0, ℓ e ): to see this, take over -with obvious changes -the proof of [38,Lemma 4.2].
Let us see why we have chosen to define an alternative inner product on L 2 (G). Under our standing assumption Q e (x)M e (x) is for all x a Hermitian matrix, so it can be diagonalized -although these matrices need not commute, so they will in general not be simultaneously diagonalizable. If however there exists a diagonal matrix D e such that For all v ∈ V, let us denote by E v , the set of all edges incident in v. We introduce for each v ∈ V the trace operator γ v : e∈E H 1 (0, ℓ e ) ke → C kv defined by where k v := e∈Ev k e , and the k v × k v block-diagonal matrix T v with k e × k e diagonal blocks (3.9) where we recall that the |V| × |E| incidence matrix I = (ι ve ) of the graph G is defined by in the first both cases, we write (3.10) v e or e v, respectively. With these notation, we see that the identity (3.8) is equivalent to Taking the real part of (3.6) and using the last identity we find The boundary terms vanish if so does γ v (u) for all v ∈ V; however, it is sufficient that this vector belongs to a totally isotropic subspace Y v associated with the quadratic form q v defined by (3.12) q in other words, a subspace of the null isotropic cone associated with this quadratic form, see Definition C.1. (Observe that q v (ξ) ∈ R, due to our standing assumptions on Q e , M e .) This means that it suffices to assume that Introducing All skew-adjoint extensions of A |D min can be then parametrized by [58,Thm. 3.6]. We are however rather interested in the general case of possibly variable coefficients and therefore prefer to pursue an approach based on the classical Lumer-Phillips Theorem. One may e.g. consider the vertex transmission conditions (see [50, p. 56

])
• p is continuous across the vertices and e∈Ev q e (v)ι ve = 0 for all v ∈ V; or • q is continuous across the vertices and e∈Ev p e (v)ι ve = 0 for all v ∈ V.
They both fit to our framework. Indeed, if for simplicity we write γ v (u) = ((p e (v)) e∈Ev , (q e (v)) e∈Ev ) ⊤ , then in the first case it suffices to take 1 Ev , ι Ev are the vectors in C Ev whose e v -entry is 1 and ι ve , respectively. On the contrary, in the second case we let Before proving our first well-posedness result, we reformulate the condition (3.13) for constant vector fields u e . Namely, if we assume that u e ≡ K e ∈ C ke for all edges e, (3.13) is equivalent to Denoting To write this in a global way, we first let k := e∈E k e . Now recall that each w (v,i) is an element of C kv , hence it can be identified with the vector (w We denote by w (v,i) ∈ C k its extension to the whole set of edges, namely, , if e ∈ E v , 0 else.
With this notation we see that (3.15), hence also (3.13) in this case, is equivalent to, In the same way each coordinate of an element of Y v , Y ⊥ v ⊂ C kv corresponds to some e ∈ E v and as above we can extend these spaces to C k by putting a 0 to the coordinate corresponding to e whenever e / ∈ E v . Denote these extensions by Y v , and Y ⊥ v , respectively.
which is further equivalent to the dimensions condition Plugging this into (3.19) yields (3.18).
Remark 3.6. The equivalent assertions in Lemma 3.5 mean that the number of boundary conditions in (3.13) (that is equivalent to (3.17) in the special case) is exactly equal to k and that these boundary conditions are linearly independent. Furthermore, as the support of the vector w (v,i) corresponds to the set of the edges incident to v, the vectors w (v,i) and w (v ′ ,i ′ ) , and hence also the We are finally in the position to formulate a well-posedness result in terms of the transmission conditions in (3.13).
Theorem 3.7. For all v ∈ V, let Y v be a totally isotropic subspace associated with the quadratic form q v defined (3.12) and assume that (3.18) holds. Then both ±A, defined on the domain are quasi-m-dissipative operators. In particular, both ±A generate a strongly continuous semigroup and hence a strongly continuous group in L 2 (G). The operator A has compact resolvent, hence pure point spectrum.
Proof. Under our assumptions, u → Nu is a bounded perturbation of A. Therefore, we can in the following assume without loss of generality that N = 0. Under this assumption and in view of (3.11) and the definition of D(A), we see that for all u ∈ D(A). By the assumptions made on the matrices Q e , M e , we deduce that for all u ∈ D(A) which means that ±A with domain D(A) are both quasi-dissipative. By the Lumer-Phillips Theorem, it remains to check their maximality. To this aim, for any f ∈ L 2 (G), we first look for a solution u ∈ D(A) of Such a solution is given by with K e ∈ C ke . It then remains to fix the vectors K e in order to enforce the condition u e ∈ D(A). Since (3.13) is in our situation a k e × k e linear system in (K e ) e∈E , the existence of this vector is equivalent to its uniqueness. By the previous considerations, this means that it suffices to show that system (3.17) has the sole solution K e = 0, which holds due to our assumption (3.18) in Lemma 3.5. This shows that the operator A is an isomorphism from D(A) into L 2 (G) and proves that ±A is maximal.
To conclude, we observe that Lemma 3.2 directly implies that A has compact resolvent, since D(A) is continuously embedded in D max .
Example 3.8. Imposing Dirichlet conditions on all endpoints is a possibility allowed for by our formalism, taking As the next result shows, this condition is easy to characterize using Lemma B.3.
Corollary 3.9. Under the assumptions of Theorem 3.7, if and only if the system (2.1) on G with transmission conditions (3.13) is governed by a unitary group on L 2 (G); in particular, the energy is conserved.
Proof. Under the assumptions of Theorem 3.7, the identity (3.11) guarantees that This obviously shows that (3.26) is a sufficient condition for the unitarity property of the semigroup generated by A. For the necessity, let us observe that Q e N e + (Q e N e ) * − (Q e M e ) ′ is hermitian.
Since the test functions vanishing at each endpoint satisfy all boundary conditions, e∈E D(0, ℓ e ) ke is included in D(A) and we deduce that (3.28) implies that for all u e ∈ D(0, ℓ e ) ke and all e ∈ E.
By Lemma B.3 we conclude that (3.26) holds. Finally, because holds along classical solutions u of (2.1), the second assertion is valid as well.
Note, that condition (3.26) is satisfied in the special case when Q e M e is spatially constant and Q e N e has zero or purely imaginary entries. such that a > 0, b ∈ C, a 2 > |b| 2 , bL ∈ R. Clearly, a different choice of Q changes the quadratic form q and hence its null isotropic cone, too, even though any two matrices Q satisfying the above assumptions are mutually similar. This shows the flexibility of our setting.
Remark 3.11. It is natural to choose transmission conditions that reflect the connectivity of the network, that is the reason of the local boundary condition (3.13). However non local boundary conditions can be imposed as well by re-writing the term v∈V T v γ v (u) · γ v (ū) in a global way as ⊤ and the 2k × 2k matrix T is given by without any reference to the structure of the network. With this notation and this suggests to replace (3.13) by i.e., to consider A with domain where Y ⊂ C 2k is a subspace of the null isotropic cone associated with the quadratic form This corresponds to glue all vertices together, thus forming a so-called flower graph, and to impose general transmission conditions in the only vertex of such a flower. This suggests to introduce the notation for any given subspace Y of C 2k . The well-posedness conditions from (3.18) in this setting sums up to This formalism also makes possible to compare solutions of the same system under different transmission conditions in the vertices. We will come back to this in the next section.
Example 3.12. The arguably easiest application of our theory is the model of flows on networks discussed e.g. in [36]. It consists of a system of k = |E| scalar equationṡ where c e are positive constants. Hence, M e = c e , and we can take Q e = 1, for all e ∈ E. The associated matrix T v is diagonal and takes the form where ι ve are the entries of the incidence matrix of the graph, see (3.9). However, in order to treat more general boundary conditions, we switch to the setting introduced in Remark 3.11 and take A rather general way of writing the transmission conditions in the vertices is [22]. They can be equivalently expressed in our formalism by The relevant conditions in Theorem 3.7 are hence whether Both conditions are e.g. satisfied if V 0 , V ℓ = Id, which corresponds to transport on k disjoint loops, in which case Theorem 3.7 confirms one's intuition that A with generates a strongly continuous group on is satisfied in the former case if c e 1 = c e 2 (hence we have by Theorem 3.7 a group generator on a network which can be regarded as a loop of lengthl ℓ e 1 + ℓ e 2 ), but not in the latter: it will follow from the results in the next section that this operator, which is a prototype of those considered in [36], still generates a strongly continuous semi group on L 2 (G). This example shows that our conditions on Y v are tailored for unitary group generation, as Corollary 3.9 shows. Some remedies to avoid such a problem will be discussed below.
We stress that the second above condition implies the invertibility of both V 0 , V ℓ , which is proved in [22,Cor. 3.8] to be equivalent to the assertion that A with domain is a group generator.

Contractive well-posedness and qualitative properties
Let us now discuss the more general situation in which the solutions to (2.1) are given by semigroups that are merely contractive. In this case, the above computations show that much more general boundary conditions can be studied. Furthermore, we are also able to describe qualitative behavior of these solutions. We refer to Section 5 for several illustrative examples.
Recall that the nonpositive isotropic cone associated with a quadratic form q : C k → R is the set of vectors ξ ∈ C k such that q(ξ) ≤ 0, see Definition C.1.
be a subspace of the nonpositive isotropic cone associated with the quadratic form q v given in (3.12)and assume that (3.18) holds. Then A with domain generates a strongly continuous quasi-contractive semigroup (e tA ) t≥0 in L 2 (G) and the system (2.1) on G with transmission conditions Furthermore, the energy in (3.27) is monotonically decreasing if (3.26) holds; in this case, there exists a projector commuting with (e tA ) t≥0 whose null space is the set of strong stability of the semigroup and whose range is the closure of the set of periodic vectors under (e tA ) t≥0 .
Proof. The proof of the assertion leading to well-posedness is exactly the same as the ones of Theorem 3.7 and Corollary 3.9 for operator A; the sufficient condition for contractivity of the semigroup can be read off (3.21).
The second assertion is a direct consequence of [ Let us continue by studying qualitative properties of the semigroup generated by A. In particular, let C be a closed and convex subset of C and write for any measure space (X, µ) observe that L 2 (X; C) is a closed and convex subset of L 2 (X). A semigroup (T (t)) t≥0 on L 2 (X) is called real (resp., positive) if each operator T (t) leaves L 2 (X; R) (resp., L 2 (X; R + )) invariant. Moreover, for a closed and convex subset K of a Hilbert space H, the minimizing projector P K onto K assigns to each u ∈ H the unique element P K u ∈ K satisfying u − P K u = min{ u − w : w ∈ K} or, equivalently (see [12,Thm. 5.2]), P K u = z ∈ K is the unique element in K such that We will use a generalization of Brezis' classical result for invariance under the semigroup generated by a subdifferential. In the linear case, [61,Thm. 2.4] can be formulated as follows.  ℜ Au, u − P K u ≤ ω u − P K u 2 for all u ∈ D(A).
If in particular ω = 0, i.e., A is dissipative, and P K u ∈ D(A) for all u ∈ D(A), then the invariance of K under (T (t)) t≥0 is equivalent to We can thus describe further properties of the semigroup generated by A in terms of the matrices Q e , M e , N e , and the boundary conditions. We are interested in the convex subsets of the form K = L 2 (G; C) where C ⊂ C is a closed interval, e.g., C = R or C = R + . Let us first relate the minimizing projector P Q K with respect to the inner product (3.2) on L 2 (G) to the minimizing projector P K with respect to the standard inner product e (x)(C) = C. Then the minimizing projector P Q K with respect to the inner product (3.2) onto K = L 2 (G; C) is given by where Q := diag(Q e ) e∈E is a block diagonal matrix and P K is the minimizing projector with respect to the standard inner product (4.5).
Proof. By (4.3), P Q K u =: z is the unique element in K such that As Q e (x) is symmetric positive definite, it admits a square root Q e (x) e w e , we may equivalently rewrite (4.8) as By (4.3) we obtain Q 1 2 z = P K v and (4.6) follows.
In many applications Q e , and thus also Q   Proof. We use Lemma 4.3 for the convex subset of real-valued functions K = L 2 (G, R) and the projector P Q K u = ℜu obtained in Equation 4.9. Also observe that we can without loss of generality assume ω = 0, since reality of a semigroup is invariant under scalar perturbations of its generator. Now, it follows from (4.10) that P Q K u ∈ D(A) whenever u ∈ D(A). We deduce that reality of the semigroup is equivalent to ℜ(Aℜu, ıℑu) ≤ 0 for all u ∈ D(A).
By applying the same trick as in the proof of [52,Prop. 2.5], that is by plugging −ℜu + iℑu into the above inequality, we obtain The conclusion then follows from Lemma 4.6 below that yeilds an equivalent, but easier to check, formulation of (4.11). Proof. As all Q e are real-valued, it suffices to show that (4.11) holds if and only if the matrices Q e M e and Q e N e are real for all e ∈ E. As this second property is clearly sufficient for (4.11) to hold, it suffices to prove the converse implication. For that purpose, fix a real-valued function ϕ ∈ D(R) with a support included into [−1, 1] and such that ϕ ′ (0) = 1. Now fix one edge e ∈ E, one point x 0 ∈ (0, ℓ e ) and one i ∈ {1, . . . , k e }. Then for all n large enough, define u n as follows: u n,e ′ = 0, for all edges e ′ = e and u n,e = ϕ(n(x − x 0 ))e i , where e i = (δ ij ) ke j=1 is the i-th vector of the canonical basis of R ke . The parameter n is chosen large enough so that the support of u n,e is included into (0, ℓ e ) so that u n ∈ D(A) (and is real valued). Taking this function in (4.11), we find By evaluating this expression at x 0 , dividing by n and and letting n goes to infinitiy, we find that In other words, Q e M e is real-valued. Once this property holds, (4.11) reduces to (Q e N e )u e is real-valued for all real-valued u ∈ D(A) and all e ∈ E.
This directly implies that Q e N e is real-valued since e∈E (D(0, ℓ e )) ke is included into D(A).
Before going on let us mention that condition (4.10) can be simplified in the following way. Proof. Let Y v be spanned by entry-wise real vectors y 1 , . . . , y n and let y ∈ Y v . Then there exist α 1 , . . . , α n ∈ C such that y = n i=1 α i y i . Because all entries of each y i are real, it follows that ℜy is again a linear combination of these basis vectors, For the converse first note that (4.10) implies that for any y ∈ Y v also ℜy, ℑy ∈ Y v . If now y 1 , . . . , y n is any basis of Y v , then the entry-wise real vectors ℜy i , Let us now continue with the study of positivity of the semigroup. Here, we will without loss of generality restrict ourselves to real Hilbert space L 2 (G, R). We first notice that each Q e (x) Lemma 4.8. Let P be a real k × k (k ≥ 1) matrix that is symmetric and positive definite. Then P is a lattice isomorphism, i.e., P (R k + ) = R k + , if and only if P is diagonal. Proof. The diagonal character of P added with its positive definiteness trivially imply that P (R k + ) = R k + . Hence, we only need to prove the converse implication. For that purpose denote by p ij , 1 ≤ i, j ≤ k (resp. q ij , 1 ≤ i, j ≤ k) the entries of P (resp. P −1 ). Now notice that from our assumption directly follows p ij , q ij ≥ 0, for all i, j. Moreover, for all i, j with i = j, we have k ℓ=1 p iℓ q ℓj = 0, or, equivalently, p iℓ q ℓj = 0 for all ℓ = 1, . . . , k.
Taking ℓ = j, we find that p ij = 0 since the diagonal entries of P −1 are strictly positive. This shows that P is diagonal as requested.
We continue by applying Lemma 4.3 to the convex subset of positive-valued functions L 2 (G, R + ). In the following we will adopt the notation  Recall also, that for any f ∈ H 1 one can write Proposition 4.9. In addition to the assumptions of Theorem 4.1, let M e , N e , Q e be real-valued for all e ∈ E, Q e (x) be diagonal for e ∈ E and all x ∈ [0, ℓ e ], and Proof. By Lemma 4.8 we may apply Lemma 4.3 to the closed convex subset K = L 2 (G, R + ). Therefore, by diagonality and positivity of Q e (x), the minimizing projector P K given by (4.6) takes the simpler form Since real scalar perturbations of the generator do not affect positivity of the semigroup, we may assume that ω = 0. As (4.12) yields P K u ∈ D(A) for all u ∈ D(A), the semigroup is thus positive if and only if ℜ(Au + , (u − u + )) ≤ 0 for all u ∈ D(A).
By applying (3.6) and (4.13) we obtain the equivalent condition The conclusion then follows from Lemma 4.11 below that furnishes an equivalent, but easier to check, formulation of (4.15).  Notice that Now fix one edge e ∈ E, one point x 0 ∈ (0, ℓ e ) and 1 ≤ i, j ≤ k e , i = j. Then for all n large enough, define v n,± as follows: v n,±,e ′ = 0, for all edges e ′ = e and v n,±,e = (v (1) n , · · · , v (ke) n ) ⊤ with only two nonzero entries: v n (x) = ϕ(n(x − x 0 )) and v (j) n (x) = −ψ ± (n(x − x 0 )). The parameter n is chosen large enough so that the support of v n is included into (0, ℓ e ) and u n,± ∈ D(A), as it vanishes in a neighborhood of each vertex. Plugging this test-function into (4.15), dividing the expression by n and letting n goes to infinitiy, we find that With the help of (4.16), we deduce that Q e (x 0 )M e (x 0 ) is diagonal.
Since this holds for all χ ∈ D(0, ℓ e ), by Lemma B.2, we conclude that (Q e (x)N e (x)) ij ≤ 0 for all i = j and all x ∈ [0, ℓ e ].  for all e, then the corresponding nonpositive isotropic subspaces Y v are isomorphic to a direct product of k v blocks of size k e that are either zero or C ke . The same holds for subspaces Y ⊥ v but with the opposite pattern. Since each edge e has exactly one initial endpoint 0 and one terminal endpoint ℓ e , we get exactly one corresponding nonzero block ( Since also γ v (u) ∈ Y v for all v ∈ V, by Theorem 4.1, A generates a strongly continuous quasicontractive semigroup.
If, the matrices Q e , M e are diagonal and real-valued, and if moreover the diagonal and offdiagonal entries of N e (x) are for all e ∈ E and a.e. x ∈ [0, ℓ e ] real and nonpositive, respectively, then the semigroup generated by A is positive. The case of spatially constant and diagonal matrices on L 1 (G) corresponds to flows in networks as studied in [36], see also [6,Chapter 18] and the references there. There, W is the generalized adjacency matrix of the line graph.
We conclude this section by elaborating on a comparison principle between semigroups. Let A 1 , A 2 be two operators, each generating a positive semigroup -say, (T 1 (t)) t≥0 , (T 2 (t)) t≥0 . Then (T 2 (t)) t≥0 is said to dominate (T 1 (t)) t≥0 if for all f ≥ 0 and all t ≥ 0.
We can now formulate the following.
satisfy our standing assumptions as well as the assumptions of Theorem 4.1. Let the semigroups generated by A 1 , A 2 , say, (T 1 (t)) t≥0 , (T 2 (t)) t≥0 , be both real and positive. Then (T 2 (t)) t≥0 dominates and additionally Proof. First of all, observe that domination is not affected if A 1 , A 2 are rescaled by the same real scalar ω; hence, we can without loss of generality assume both A 1 , A 2 to be dissipative. The proof can then be performed combining Lemma 4.3 and [52,Thm. 2.24].
An interesting case arises when A 1 , A 2 are defined by means of the same matrices M, Q, N, and only the boundary conditions are different, i.e., Y 1 = Y 2 . This is e.g. the case if Y 1 is the space spanned by Πv 1 , . . . , Πv m , where v 1 , . . . , v m are the vectors spanning Y 2 and the projector Π is a block diagonal operator matrix whose diagonal blocks are a zero matrix (of any size ≥ 1) and an identity matrix (of any size ≤ 2|E|). In the case of the transport equation on a network studied in [36], this corresponds to comparing a given network with a new network with additional Dirichlet conditions in some vertices.
We may discuss in a similar way the issue of L ∞ -contractivity, i.e., the invariance of L 2 (G; C) for C := [−1, 1] under the semigroup generated by A. We omit the details.

Linearized Saint-Venant models.
Here we study a system where the linearized Saint-Venant model (2.6) is considered on all the J edges of a network, and hence the unknown (h, u) ⊤ is replaced by (h e , u e ) ⊤ e∈E . While our approach applies to arbitrary networks and variable functions H, V that may differ across the edges, for the sake of simplicity we here restrict to the case of constant (real) coefficients H, V that are independent of the edges, to H > 0 (which is physically reasonable), and to a star-shaped network. More precisely, for some integer J ≥ 2 we let E := {e 1 , . . . , e J }, and identify each edge e j with (0, ℓ j ): v 0 will correspond to the endpoint ℓ 1 for e 1 and to the endpoint 0 for all other edges. The external vertex of e j will be denoted by v j , see  Now we need to fix the boundary conditions at all vertices. According to our approach they are related to the operators T v defined in (3.9). In our case we have (see (2.8)) Because the matrix B is independent of e, and also symmetric and invertible due to condition (2.7), we thus have T v 1 = B, and T v i = −B for all i ≥ 2 at the external vertices v i , i ≥ 1, while at the interior vertex v 0 T v 0 is a block diagonal matrix: Now, let us notice that the two eigenvalues λ ± of B satisfy Hence, under the subcritical flow condition gH − V 2 > 0 (see [5, p. 14]), λ + and λ − are of opposite sign. On the contrary, under the supercritical flow condition gH − V 2 < 0, λ + and λ − have the same sign; but as λ + + λ − = (g + H)V > 0, in this case they are both strictly positive. Now we distinguish between these two flow conditions. 1) If gH − V 2 < 0, then at the external vertices the only choice for a totally isotropic subspace This already yields 2J boundary conditions and there is no more freedom to manage the internal vertex v 0 . In other words, under this choice of boundary conditions the associated operator A cannot generate a group. But we may hope for the generation of a semigroup. Hence as T v 1 = B has two positive eigenvalues we surely need to impose that Y v 1 = {(0, 0) ⊤ }, i.e., while we are free to impose boundary conditions or not at v j , for all j ≥ 2. Since T v 0 has 2 positive eigenvalues and 2(J − 1) negative eigenvalues, by Lemma C.3, the maximal dimension of a subspace Y v 0 of the nonpositive isotropic cone associated with T v 0 is 2(J − 1). Choosing Y v 0 as the subspace associated with the negative eigenvalues leads to a decoupled system and is of less interest. Letting instead 0, 1, 0, . . . , 1, 0) ⊤ , (0, 1, 0, 1, . . . , 0, 1) ⊤ }, 0, 1, 0, . . . , 1, 0) ⊤ + y(0, 1, 0, 1, . . . , 0, 1) ⊤ for some x, y ∈ C, since B is positive definite. This choice corresponds to the continuity of the water depth and the velocity at v 0 , namely and yields 2(J − 1) boundary conditions. They are complemented by the two conditions (5.1) at v 1 , and by no conditions at v i for i ≥ 2. This leads to k = 2J boundary conditions for which To conclude the generation of a semigroup by Theorem 4.1 it suffices to notice that Furthermore, by Lemma 4.6 and Proposition 4.5, the semigroup is real. Finally, by Lemma 4.11 condition (4.15) does not hold, hence the semigroup is not positive.
2) If we are in the subcritical case gH − V 2 > 0, then the eigenvalue λ + (resp. λ − ) is positive (resp. negative). Let us first analyze the possibility to have a group. In that case, by Lemma C.2, at any external vertex v a totally isotropic subspace of q v is of dimension at most one, while at the interior vertex v 0 it is at most J. Let us present the following example. If U ± is the normalized eigenvector of B associated with λ ± , then according to (C.8), is an isotropic vector of the sesquilinear form associated with B. Therefore for all j = 1, . . . , J, we take Y v j as the vector space spanned by U. We proceed similarly at v 0 by fixing J isotropic vectors constructed in the proof of Lemma C.2. To have a coupling system, one possibility is the following one. We notice that the eigenvectors of T v 0 associated with positive eigenvalues are U + 1 = (U ⊤ + , 0, . . . , 0) ⊤ for the eigenvalue λ + and . . , J which is also an independent set itself, and (3.18) holds since v∈V dim Y v = 2J. Furthermore, each Y v is by construction a totally isotropic subspace associated with the quadratic form q v , hence we are in the position to apply Theorem 3.7 and deduce that the associated operator A generates a group, which is by Corollary 3.9 is unitary if and only if V = 0. As before the (forward) semigroup is real and does not preserve positivity.
In the subcritical case gH − V 2 > 0, examples of boundary conditions leading to a semigroup can be easily built as before.

Wave type equations.
Wave-type equations on graphs have been intensively studied in the literature, let us mention [2,33,39,42,48,51]. Here we focus on extending these results to rather general elastic systems modeled as where α e ∈ C 1 ([0, ℓ e ]) and β e , γ e ∈ L ∞ (0, ℓ e ) are real-valued functions. For the sake of simplicity, we hence restrict to stars as in Figure 5.1, which can be regarded as building blocks of more general networks. It turns out that (5.2) is equivalent tȯ U e = M e U ′ e + N e U e , for the vector function U e = (u ′ e ,u e ) ⊤ , where As M e is symmetric, Assumptions 2.1 are automatically satisfied by choosing Q e as the identity matrix. As usual, the boundary conditions at the vertices are related to the values of M e at the endpoints of the edge e, that generically are given by when v is one of the endpoints of e; hence M e (v) has two real eigenvalues of opposite sign, We are thus in the same situation as in the Saint-Venant model with the subcritical condition: we thus do not give any further details about the choice of boundary conditions at the interior vertices, since all ideas presented there carry over to the present case. Note that for an exterior vertex v, Neumann condition u ′ e (v) = 0, can be equivalently described by means of the totally isotropic subspace Y v spanned by (0, 1) ⊤ . In this case, (0, 1) ⊤ will be an isotropic vector if and only if α e (v) = 0. On the contrary, Dirichlet boundary condition at an exterior vertex v u e (v) = 0, leads tou e (v) = 0, and corresponds to the choice Y v spanned by (0, 1) ⊤ . Our approach also allows us to discuss absorbing boundary condition (see [17] for instance)) u ′ e (ℓ e ) = −κu e (ℓ e ) with κ ∈ (0, ∞): indeed it then corresponds to the space spanned by U = (−κ, 1) ⊤ , hence that will be nonpositive as soon as κ ≥ α e (v)/2.
Let us finally notice that provided the boundary conditions are nice enough to generate a group, it will be unitary group if and only if γ e = 0 and β e = α ′ e 2 , for every edge e. In case of a generation of a semigroup, it will be real provided (4.10) holds, while if all γ e are nonpositive, it will be never positive (this is in particular the case for the wave equation, as α e = β e = γ e = 0).

5.3.
Hybrid transport/string equations. Network-like systems described by equations which are partially of diffusive and partially of transport type have been studied by Hussein and one of the authors in [27]; characterization of the right transmission conditions leading to well-posedness has proved a difficult task. In the following we turn to the different but related task of connecting transport and wave equations; this can suggest natural ways of coupling first and second order differential operators. The simplest toy model is to consider a scalar wave equation set in an interval (0, ℓ 2 ) and a scalar transport equation in (0, ℓ 1 ) coupled via their common endpoint that is assumed to be 0. This means that we consider the system with Dirichlet boundary condition at ℓ 2 where α, β are two real numbers fixed below in order to guarantee well-posedness of our system.
As in Subsection 5.2, by identifying the interval (0, ℓ i ) with e i , i = 1, 2, introducing u e 1 = (u ′ ,u) ⊤ = (u e 1 ,1 , u e 1 ,2 ) ⊤ , and setting u e 2 = p, we can transform our system into a first order system of the form (2.1) with M e 1 = 0 1 1 0 and therefore Q e 1 = 1 0 0 1 and M e 2 = −1 (hence Q e 2 = 1). Notice that our network possesses three vertices, v 0 , corresponding to 0 (with E v 0 = {e 1 , e 2 }), v 1 ,corresponding to the endpoint ℓ 1 of e 1 (with E v 1 = {e 1 }), and v 2 , corresponding to the endpoint ℓ 2 of e 2 (with E v 2 = {e 2 }). Since there is no boundary condition at v 1 , we set Y v 1 = C 2 , to take into account (5.4), we set Y v 2 = C × {0}, and (5.5) requires to take With this notation, we see that This means that Y v 0 is a subspace of the nonpositive isotropic cone associated with q v 0 if and only if 2αβ ≥ 1.
Here we have two interior vertices v 1 , v 2 : the former corresponds to the endpoint 0 of both (0, ℓ 1 ) and (0, ℓ 2 ), while the latter corresponds to the endpoint 0 of (0, ℓ 3 ) and the endpoint ℓ 1 of (0, ℓ 1 ). Hence imposing conditions (5.4)-(5.5) as before along with Dirichlet boundary condition at ℓ 3 in the case of the string-like and beam-like edges, respectively. In this case, Theorem 4.1 applies and we deduce that the hyperbolic system is governed by a strongly continuous semigroup. Indeed, arguing as in Remark 3.11 we deduce that this semigroup is contractive, hence the energy of solutions is decreasing. The aim in [3] was to discuss the stabilization of an elastic system: the reason why it makes sense to consider closed feedbacks is that if they are replaced by zero conditions, then in view of the duality between continuity and Kirchhoff conditions, a direct computation shows the assumptions of Corollary 3.9 are satisfied and we conclude that the system is governed by a unitary group. 5.5. The Dirac equation. The 1D Dirac equation is briefly discussed in [59, § 1.1]: it was later extended to the case of networks and thoroughly studied by Bolte and his coauthors [8,9], who also observed that it then takes on each edge the form for a C 2 -valued unknown ψ = (ψ (1) , ψ (2) ). A parametrization of skew-adjoint realizations on a network has been presented in [8]; we are going to study the more general problem of finding boundary conditions that lead to a group or merely a semigroup, which still yields forward wellposedness of the Dirac equation.
To begin with, observe that Assumptions 2.1 are especially satisfied by taking however, see also Example 3.10. We hence deduce from Lemma 4.6 that, no matter what the boundary/transmission conditions look like, a semigroup governing the Dirac equation cannot be real, let alone positive. Let us now study the quadratic form q v . We switch to the global viewpoint presented in Remark 3.11 (i.e., we allow for interactions that are nonlocal with respect to the given underlying network) and observe that by (3.9), T is a diagonal 2|E| × 2|E| block matrix whose first and last |E| diagonal blocks are 0 −ıc ıc 0 and 0 ıc −ıc 0 , respectively. Therefore, (ξ, η) = (ξ 1 , η 1 , . . . , ξ 2|E| , η 2|E| ) ∈ C 4|E| is an isotropic vector for the quadratic form q if and only if Any 2|E|-dimensional space Y of vectors satisfying (5.8) induces boundary conditions that determine a realization of A generating a group; in fact, necessarily a unitary group, since (3.26) is clearly satisfied. A well-known example is that of vectors ξ being scalar multipliers of the "characteristic function" 1 Ev := 1 {e∈Ev} and η with the same support E v and orthogonal to ξ, where E v ⊂ E is the set of edges incident with any given vertex v; this gives rise to Y = v∈V Y v with corresponding to continuity of the first coordinate of ψ across all vertices, and a Kirchhoff condition on the second coordinate; clearly, swapping the transmission conditions in vertices satisfied by the two coordinates yields again a group generator. Another possibility is e.g. given by letting Y : e (0) and ψ (1) e (ℓ e ) = ψ (2) e (ℓ e ). If we turn to the issue of mere contractive well-posedness, then we observe that any non positive isotropic cone consists of vectors such that accordingly, by Theorem 4.1 any 2|E|-dimensional Y all of whose elements satisfy (5.9) induces a realization of the operator A that generates a strongly-continuous contractive semigroup. A somewhat trivial example is given by Y := {0}⊕C 2|E| , corresponding to decoupled case of Dirichlet boundary conditions imposed at the left endpoint of each interval on both coordinates of the unknown; or, more generally (and interestingly) of Y = {(ıBη, η) : η ∈ C 2|E| } for some accretive (but not necessarily Hermitian) matrix B of size 2|E|. 5.6. Second sound in networks. So-called "second sound" is an exotic, wave-like phenomenon of heat diffusion that was first proposed by Landau to explain unusual behaviors in ultracold helium. Second sound has ever since been observed in several materials -most recently by Huberman et al. [26] also in graphite around cozy 130 • K. As thoroughly discussed in [55], one classical model going back to Lord and Shulman [43] boils down to the linear equations of thermoelasticity where z, θ, and q represent the displacement, the temperature difference to a fixed reference temperature, and the heat flux, respectively, and α, β, γ, δ, τ 0 , κ are positive constants. Racke has discussed in [55] the asymptotic stability of this system under three classes of boundary conditions: proving in detail well-posedness in the case of (i) and suggesting to use a similar strategy to study (ii) and (iii). In fact, the boundary conditions (iii) actually represents a dynamic condition for the unknown q at 0, and hence seem to require a subtler analysis: we will consider them along with further hyperbolic systems with dynamic boundary conditions in a forthcoming paper [35]. We rewrite ( The choice of Q e is rather natural and indeed a similar term was also used to regularize the inner product by Racke, see [55, (18)]. A direct computation shows that with four eigenvalues of the form ± H±2 , where H := α 2 δ 2 + β 2 δ 2 + β 2 γ 2 and K := H 2 − 4α 2 β 2 γ 2 δ 2 . Because H 2 > K whenever α, β, γ, δ > 0, Q e M e has two positive and two negative eigenvalues. This is coherent with both above choices of boundary conditions (in the purely hyperbolic case of τ 0 = 0). For the sake of simplicity, let us focus for a while on the case of an individual interval that can be expressed in our formalism taking as Y at each endpoint the spaces respectively. A further possible choice for a subspace of the null isotropic cone is e.g.
This paves the way to the study of second sound on collection of intervals with coupled boundary conditions, an especially interesting issue, as second sound has been conjectured in [26] to take place in graphene -a network: more precisely, hexagonal lattice of carbon atoms -, already at room temperature.
Indeed, one can apply our general theory in order to describe transmission conditions leading to well-posedness; an easy computation shows that the relevant equation is a higher dimensional counterpart of (5.11). An educated guess suggests to study conditions of continuity (across the ramification nodes) on both displacement and temperature, i.e., on z -henceż -and θ, along with a Kirchhoff-type condition on z ′ and q. It is remarkable that this choice does not satisfy (5.11). However, it is not difficult to see that all boundary values that satisfy either on z -henceż -as well as q define a totally isotropic subspace of the null isotropic cone. (If the vertex v has degree 1, then in both cases the first conditions become void, whereas the second reduce to Dirichlet conditions.) Again, we see that Equation 3.18 is satisfied and conclude that the system is governed by a strongly continuous group on L 2 (G). All above spaces Y are invariant under taking both the real and the positive part. Furthermore, Q, M, N are real valued and Q, N are diagonal, but M is not, hence by Proposition 4.5 and Proposition 4.9 the semigroup generated by A with any of these transmission conditions is real but not positive.
Furthermore, A generates merely a semigroup whenever the space Y defining the boundary conditions is a subspace of the nonpositive isotropic cone of T : this can e.g. enforced by assuming that Z 1 = BZ 2 and (Z 2 + Q) = −CΘ for some dissipative matrices B, C, provided Y has the correct dimension. Because Finally as a composition of Lipschitz continuous mappings, Q e (·) is Lipschitz continuous, too.
Conversely, let us assume that Assumptions 2.  We first prove a density result in the subset of positive integrable functions; we recall the notation in (4.2) and write likewise D(0, ℓ; R) for the set of real-valued test functions. Proof. Indeed let us fix u ∈ L 1 (0, ℓ; R + ), then √ u belongs to L 2 (0, ℓ; R) and therefore there exists a sequence (ϕ n ) n∈N of functions in D(0, ℓ; R) such that Therefore by Cauchy-Schwarz's inequality we have By (B.1), we conclude that this right-hand side tends to zero. Now we prove two variants of the fundamental lemma of the calculus of variations (or "du Bois-Reymond's lemma"). Again A ij + A ji = 2ℜA ij is real-valued because A is hermitian, and Lemma B.2 yields ℜA ij = 0.
Appendix C. On subspaces of isotropic cones associated with a quadratic form In this section we fix a positive integer k and a hermitian and invertible matrix P ∈ C k×k . Its associated quadratic form q is defined by q(ξ) = P ξ ·ξ, ξ ∈ C k . Now we introduce some cones associated with q, see [40,Def. 3.1].
Definition C.1. 1) The null isotropic cone associated with the quadratic form q is defined as the set of isotropic vectors associated with q, namely the set of vectors ξ ∈ C k such that (C.1) q(ξ) = 0.
A subspace of the null isotropic cone associated with q is called a totally isotropic subspace and the isotropy index (of the quadratic space associated with q), denoted here by i(q), is the maximum of the dimensions of the totally isotropic subspaces.
2) The nonpositive isotropic cone associated with the quadratic form q is defined as the set of vectors ξ ∈ C k such that (C.2) q(ξ) ≤ 0.
From Lemma 1.2 of [47] we know that i(q) ≤ k/2 but, surprisingly, we could not find in the literature a reference that yields a characterization of i(q). Hence the goal of this appendix is to characterize this isotropic index as well as the maximal dimension of any subspace of nonpositive isotropic cones.
Let {λ i } k i=1 be the set of eigenvalues of P , repeated according to their multiplicities and enumerated in an increasing order, and denote by {u i } k i=1 the set of the associated normalized eigenvectors, i.e., P u i = λ i u i and u i ·ū j = δ ij for all i ∈ {1, . . . , k}.
Denote by k − (resp. k + ) the number of negative (resp. positive) eigenvalues of P . Without loss of generality, we may assume that λ i < 0 for i ≤ k − and λ i > 0 for i > k − . We will see that the isotropic index i(q) agrees with min{k − , k + }.
Lemma C.2. Any subspace of the null isotropic cone associated with the form q has dimension at most κ := min{k − , k + }. Furthermore if κ ≥ 1, there exist at least 2 κ subspaces of the null anisotropic cone associated with q of dimension κ.
Proof. If κ = 0, this means that P is either positive definite or negative definite and therefore the associated isotropic cone is reduced to {0}. So the only case of interest is the case κ ≥ 1. By symmetry, we can assume that κ = k − . So let us now fix a subspace I of the isotropic cone associated with q. Every nonzero u ∈ I can be written as for some α i ∈ C which are not all zero. Since (C.1) is equivalent to we find that there exists at least one i ≤ k − such that α i = 0. Assume that K := dim I > k − and let {U i } K i=1 be a basis of I. Let us write for some α ij ∈ C. By the previous remark, for all i ≤ K, there exists j ≤ k − such that α ij = 0. Now we use a sort of Gram-Schmidt procedure: Starting with i = 1 and without loss of generality (else we change the enumeration) we can assume that α 11 = 0 and, consequently, we have where we have setŨ 1 := U 1 and δ 1 := 1 α 11 . Plugging this expression into (C.5) with i = 2, we find that (C.6) U 2 = α 21 α 11 U 1 + k j=2α 2j u j , with someα 2j ∈ C. This means that the new vectorŨ 2 := U 2 − α 21 α 11 U 1 , that is still in I, has at least one coefficientα 2j different from zero for j ∈ {2, . . . , k − }. Again, after a possible change of enumeration, we can assume thatα 22 = 0, hence we have where δ 2 = 1 α 22 . Note that the new set {Ũ 1 ,Ũ 2 } ∪ {U i } K i=3 forms a basis of I. By iterating this procedure, after k − steps, we will find a basis β ij u j for all i = 1, . . . , k − , for some δ j ∈ C\{0} and some β ij ∈ C. By using the expansion (C.5) of U k − +1 and (C.7), we find for some γ ij ∈ C * . We then arrive to a contradiction because on one hand the vector V := U k − +1 − k − j=1 α ij δ jŨj is in I, hence q(V ) = 0, while on the other hand V = 0 is a linear combination of the u j 's for j ≥ k − + 1, hence q(V ) > 0.
For the last assertion, if in (C.3), for all i = 1, . . . , k − , we chose α i = 1 and α i ′ = 0 for all i ′ / ∈ {i, k − + i}, condition (C.4) will hold if and only if or equivalently This yields the isotropic vectors And, since the U + i 's and the U − i 's are linearly independent, we find 2 k − possibilities.
A similar assertion holds for subspaces of the nonpositive isotropic cone associated with the quadratic form q -one such subspace is spanned by the first k − eigenvectors {u i } k − i=1 of P . Lemma C.3. Any subspace of the nonpositive isotropic subspace associated with the form q has dimension at most k − .
Proof. If k − = 0, this means that P is positive definite and therefore the only possible choice for such a subspace is {0}. So the only case of interest is the case k − ≥ 1. Now the proof is exactly the same as the one of the previous Lemma. Indeed let I be such a subspace and let u ∈ I different from zero, then it admits the splitting (C.3) with some α i ∈ C not all zeroes. Since the constraint P u ·ū ≤ 0, is equivalent to we again find that there exists at least one i ≤ k − such that α i = 0. The previous argument then leads to dim I ≤ k − .