Asymptotics and estimates for spectral minimal partitions of metric graphs
Matthias Hofmann, James B. Kennedy, Delio Mugnolo, Marvin Plümer
aa r X i v : . [ m a t h - ph ] J u l ASYMPTOTICS AND ESTIMATES FOR SPECTRAL MINIMAL PARTITIONSOF METRIC GRAPHS
MATTHIAS HOFMANN, JAMES B. KENNEDY, DELIO MUGNOLO, AND MARVIN PL ¨UMER
Abstract.
We study properties of spectral minimal partitions of metric graphs within the frame-work recently introduced in [Kennedy et al (2020), arXiv:2005.01126]. We provide sharp lower andupper estimates for minimal partition energies in different classes of partitions; while the lowerbounds are reminiscent of the classic isoperimetric inequalities for metric graphs, the upper boundsare more involved and mirror the combinatorial structure of the metric graph as well. Combin-ing them, we deduce that these spectral minimal energies also satisfy a Weyl-type asymptotic lawsimilar to the well-known one for eigenvalues of quantum graph Laplacians with various vertexconditions. Drawing on two examples we show that in general no second term in the asymptoticexpansion for minimal partition energies can exist, but also that various kinds of behaviour arepossible. We also study certain aspects of the asymptotic behaviour of the minimal partitionsthemselves. Introduction
Spectral minimal partitions were first introduced on planar domains in [CTV05] and havebeen a popular topic within spectral theory ever since; we refer the interested reader to the sur-vey [BNH17]. As such spectral minimal partitions represent a natural way to partition an objectwhich reflects both its geometric and its metric structure, it is natural to study them on metricgraphs.In a recent joint work with Pavel Kurasov and Corentin L´ena [KKLM20], two of the presentauthors undertook what was perhaps the first systematic study of such partitions on metric graphs,which had first been considered in a rather different context in [BBRS12]. Roughly speaking,associated with each partition P of a metric graph G into k connected subgraphs G , . . . , G k ,or clusters , one can consider the p -mean Λ p ( P ), p ∈ [1 , ∞ ], of the k -vector of lowest positiveeigenvalues of a suitable Laplacian restricted to each such G i ; and then minimise Λ p ( P ) over all k -partitions P , leading to some number L k,p ( G ); the spectral minimal partitions are the partitionsattaining this minimal value. It should be stressed that different choices of vertex conditions(roughly speaking, of Dirichlet or Neumann type) to be imposed at the cut points of the partition,as well as different regularity assumptions on the admissible partitions, lead to different notions ofspectral minimal partitions and associated spectral minimal energies. (See Section 2 for a summaryof the key definitions.)The construction and well-posedness of such spectral problems was discussed in [KKLM20]; moreprecisely, it was shown there that the existence of (various notions of) spectral minimal partitions Mathematics Subject Classification.
Key words and phrases.
Quantum graphs, Spectral minimal partitions, Weyl asymptotics, Spectral geometry.The authors would like to thank Pavel Kurasov and Jiˇr´ı Lipovsk´y for helpful comments on the Weyl asymp-totics of quantum graphs. The work of M.H. and J.B.K. was supported by the Funda¸c˜ao para a Ciˆencia e aTecnologia, Portugal, via the program “Investigador FCT”, reference IF/01461/2015 (J.B.K.), and via projectPTDC/MAT-CAL/4334/2014 (M.H. and J.B.K.). The work of D.M. and M.P. was supported by the DeutscheForschungsgemeinschaft (Grant 397230547). All the authors would like to acknowledge networking support by theCOST Action CA18232. is equivalent to the existence of a minimum of certain energy functionals: their critical points canin turn be studied by classical variational methods, and exist on all finite metric graphs G and withrespect of several different notions of “partition”. Several qualitative properties of such partitionswere also discussed; in particular, the close relationship between the minimal partitions and theirenergies on the one hand, and eigenfunctions and eigenvalues of the Laplacian on the same metricgraph on the other, was considered in some detail in [KKLM20, § § k -partitions may be reasonableproxies for the eigenvalues of the Laplacian with natural vertex conditions, in addition to theirinterpretation as a way to partition a graph into k “analytically similar” pieces. It is thus natural toconsider their qualitative and quantitative properties, in particular in terms of how these spectralminimal energies depend on “geometric quantities” of metric graphs like the total length or thenumber of vertices of degree one, as well as their asymptotic behaviour for large k , a question ofsome interest on domains (see the discussion in [BNH17, § § k andasymptotically sharp for each graph as k → ∞ , we can obtain asymptotic relations of Weyl typefor the spectral minimal energies which strongly recall the eigenvalue Weyl asymptotics. Moreprecisely, we will show that the energies L k,p grow as(1.1) π L k + O ( k ) as k → ∞ , exactly like the eigenvalues of various realisations of the Laplacian on compact metric graphs[Nic87, CW05, BE09, OS19]. This may also be compared with the case of planar domains, wherea two-dimensional analogue of (1.1) is conjectured for the (Dirichlet) spectral minimal energies –the so-called hexagonal conjecture – but to date only two-sided asymptotic bounds are available(see [BNH17, § L k,p , the asymptotics (1.1), and also a result on the asymptoticbehaviour of the optimal partitions themselves, are collected in Section 3. The lower bounds on SYMPTOTICS AND ESTIMATES FOR SPECTRAL MINIMAL PARTITIONS OF METRIC GRAPHS 3 L k,p require different techniques from the upper bounds. Hence we group all the proofs of the lowerbounds, including extensions of the results of Section 3, in Section 4 and the proofs and extensionsof the upper bounds in Section 5; in each section, we first discuss the case of Dirichlet partitions,which is more instructive and sometimes more delicate, and then give the corresponding results inthe natural case. Section 6 is devoted to the proof of our main result on the asymptotic behaviourof the spectral minimal partitions (Theorem 3.3), which controls the size of the maximal clusterwithin the partition, as well as certain consequences of this result (see Section 3 for details).Note that there is no general known inequality between the lowest non-zero eigenvalues of theLaplacian with Dirichlet and with natural vertex conditions. Hence it seems that, likewise, no im-mediate inequality between spectral minimal energies with Dirichlet and natural vertex conditionsis available. In particular, this means as yet no interlacing techniques are available, e.g. for thepurpose of proving asymptotics of energies of spectral minimal k -partitions as k → ∞ : all ourresults have to be proved separately for the cases of Dirichlet and natural conditions.We conclude our note by discussing, in Section 7, two simple illustrative examples which allowus to show the rich behaviour of the correction term O ( k ) in (1.1) and in particular show that thecorrection term O ( k ) in (1.1) will not in general contain any first order term. Rather, if we write L k,p = π L k + c k k , c k ∈ R , then the set of points of accumulation of the c k may be a finite setof any cardinality, as our first example of equilateral stars shows (Section 7.1), or an interval, asour second example of two disjoint intervals of incommensurate lengths shows (Section 7.2). Wealso set up the former to give an explicit example for which there is no second term in the Weylasymptotics at the same time, see Remark 7.3. It seems reasonable to expect these two typesof behaviour to be generic for graphs with rationally dependent and independent edge lengths,respectively, and the same to hold for the Weyl asymptotics of the Laplacian eigenvalues, but itwould go well beyond the scope of this note to explore the question further.For ease of reference, we collect some useful isoperimetric-type inequalities for Laplacian eigen-values on graphs in an appendix.2. Finite quantum graphs and spectral minimal partitions
As mentioned, the present paper is strongly motivated by the setting introduced in [KKLM20],which we are going to recall and summarise briefly (and every now and then somewhat imprecisely)for the convenience of the reader. We start with some preliminaries on metric graphs.A compact metric graph G is a finite disjoint union of bounded intervals ( I e ) e ∈ E connected ina network-like fashion by possibly gluing their endpoints. The set V of glued end points will bereferred to as the vertex set of G ; the set E will be refered to as the edge set of G . The length of anedge e ∈ E – which is the length of the corresponding interval I e – will be denoted by | e | . Given ameasurable subset A ⊂ G its Lebesgue measure will be denoted by |A| ; note that this notation isin line with the notation for the edge lengths in G , as each edge e ∈ E will be identified with thesubset of G corresponding to the interval I e . The total length of G will be denoted by L := |G| = X e ∈ E | e | . We say G is connected if it is connected as a metric space for the canonical distance functioninduced by Euclidean distance on each edge and the above construction. We refer to [Mug19] fora more rigorous definition of G as a metric measure space and the function spaces C ( G ), L ( G ),and H ( G ) defined on it. Now, given a subset V of V , H ( G ; V ) is the ideal of H ( G ) consistingof all H ( G )-functions vanishing at V . M. HOFMANN, J. B. KENNEDY, D. MUGNOLO, AND M. PL ¨UMER
Given a metric subgraph H of G , we can consider the quadratic Dirichlet form a ( f ) := ˆ H | f ′ ( x ) | d x on the domain H ( H ) or H ( H ; V ), the latter for a given set V of vertices in H . In the formercase, the associated operator is the Laplacian with so-called standard or natural vertex conditions;the functions in its domain are in C ( H ) ∩ L ( H ) and are edgewise H , and satisfy a Kirchhoffcondition; in the second case, the functions additionally satisfy a Dirichlet (zero) condition atevery vertex in V . Such Laplacians defined on metric graphs are usually known as quantumgraphs . Since H ⊆ G is a compact metric graph, such Laplacians are self-adjoint operators withcompact resolvent and in particular have discrete, real spectrum. We will be interested in theirrespective smallest nontrivial eigenvalues, which may be described variationally by(2.1) µ ( H ) = inf (cid:26) ´ H | f ′ ( x ) | d x ´ H | f ( x ) | d x : 0 = f ∈ H ( H ) and ˆ H f ( x ) d x = 0 (cid:27) in the case of the Laplacian with standard vertex conditions, and(2.2) λ ( H ) = inf (cid:26) ´ H | f ′ ( x ) | d x ´ H | f ( x ) | d x : 0 = f ∈ H ( H ; V ) (cid:27) for the Laplacian with at least one Dirichlet condition, i.e., if V = ∅ . Equality in each case isachieved exactly when f is a corresponding eigenfunction. These eigenvalues may be shown to bestrictly positive if H is connected.We will actually consider families of Laplacians and eigenvalue problems, each defined on theclusters of a partition of G ; such a partition is by definition a family P = {G , . . . , G k } of connected,distinct, metric subgraphs of G (its clusters ) with mutually disjoint interiors and whose union yields G .In [KKLM20], no fewer than five notions of partitions are introduced, which satisfy differingconditions on the behaviour of distance function dist on G , its restrictions to the clusters G i , andtheir behaviour close to the cut points (i.e., to the points that are cut through to generate theclusters). In this paper, we are going to deal with only two of them: rigid and loose partitions, theonly sets of partitions that are closed with respect to the arguably natural topology on the spaceof all partitions discussed in detail in [KKLM20, § §
2] for details; we can summarise the ideas behindtheir definition as follows: • Loose partitions allow for clusters G i that arise by possibly cutting through any of theirpoints, as long as each G i is still connected. • Rigid partitions are more restrictive in that they only admit clusters that arise by cuttingexclusively through the separating points , i.e., those points in G on the boundary betweenclusters, although these points may be cut arbitrarily as long as the G i are still connected.The set of all rigid partitions is thus a subset of the set of all loose partitions of G . The differencebetween these notions is probably best explained by the example, given in Figures 2.1 and 2.2of a lasso graph, with two rigid (and hence also loose) partitions, and one loose (but non-rigid)partition. SYMPTOTICS AND ESTIMATES FOR SPECTRAL MINIMAL PARTITIONS OF METRIC GRAPHS 5 vw ze e e Figure 2.1.
The lasso G . w v vv ze e e vvw ze e e w v v zze e e Figure 2.2.
Left and centre: two different rigid 2-partitions of G (the only sepa-rating point is v ); right: a loose 2-partition of G (the only separating point is v butwe are additionally cutting through z ; observe that additionally cutting through v would not be admissible, as this would yield a 3-partition).Because G is a compact metric graph, each cluster G i is compact and connected; in particular, λ ( G i ) and µ ( G i ) are well defined and strictly positive, where in the former case the Dirichletconditions are taken at the cut points of G i .Given a k -partition P = {G , . . . , G k } we can hence consider its energies (2.3) Λ Np ( P ) = (cid:18) k k P i =1 µ ( G i ) p (cid:19) /p if p ∈ [1 , ∞ ) , max i =1 ,...,k µ ( G i ) if p = ∞ , and(2.4) Λ Dp ( P ) = (cid:18) k k P i =1 λ ( G i ) p (cid:19) /p if p ∈ [1 , ∞ ) , max i =1 ,...,k λ ( G i ) if p = ∞ , respectively. We can finally consider L Nk,p ( G ) and e L Nk,p ( G ) , the minima of Λ Np ( P ) over all rigid/loose k -partitions, respectively; and L Dk,p ( G ) , the minima of Λ Dp ( P ) over all rigid k -partitions. It was proved in [KKLM20, Corollary 4.8] thatunder our assumptions on G all these problems do indeed admit minima (by [KKLM20, Lemma 4.3],it is pointless to study the minimum Λ Dp ( P ) over all loose k -partitions, as it always agrees with L Dk,p ( P ).) We generically refer to all these minima as spectral minimal energies of G and, asmentioned, the corresponding minimising partitions as spectral minimal partitions .Finally, we introduce a number of quantities of a given graph G which will be important in thesequel. Definition 2.1.
Let G be a compact, connected metric graph. M. HOFMANN, J. B. KENNEDY, D. MUGNOLO, AND M. PL ¨UMER (1) The longest edge length of G will be denoted by ℓ max := max e ∈ E | e | ; the shortest edge lengthby ℓ min := min e ∈ E | e | .(2) The Betti number β ≥ G is the number of independent cycles in G ; equivalently, β = | E | − | V | + 1.(3) The girth s ∈ (0 , ∞ ] is the length of the shortest cycle in G ; if G is a tree, then it is definedto be ∞ .3. Main results: asymptotic behaviour of the optimal energies and partitions
We start by summarising our principal results, which give concrete two-sided bounds on thequantities L Dk,p ( G ), L Nk,p ( G ) and e L Nk,p ( G ), and as a consequence describe their asymptotic behaviour.Actually, we can say more, both about the asymptotic behaviour of the clusters of the optimalpartitions, and in terms of concrete two-sided bounds on these quantities for finite k . The compact,connected metric graph G will be fixed throughout, and we recall, in addition to the notation fromDefinition 2.1, that G is taken to have | E | ≥ L , and | N | vertices of degree one. Theorem 3.1.
Let p ∈ [1 , ∞ ] . Then π kL (cid:0) k + 3( k − β − | N | ) (cid:1) ≤ L Dk,p ( G ) ≤ π L (cid:18) k + (cid:18) | E | − − (cid:22) | N | (cid:23)(cid:19)(cid:19) for all sufficiently large k ≥ , in particular for k ≥ max (cid:26) β + | N | , Lℓ min + | E | − (cid:27) . In particular, (3.1) L Dk,p ( G ) = π L k + O ( k ) as k → ∞ . This theorem will be an immediate consequence of the results of Sections 4.1 and 5.1; see inparticular Corollary 4.6 and Theorem 5.1. Actually, we can give slightly sharper (but often moreinvolved) lower bounds in some cases; see Theorems 4.1 and 4.5 and Corollary 6.1.
Theorem 3.2.
Let p ∈ [1 , ∞ ] . Then (3.2) π L k ≤ e L Nk,p ( G ) ≤ L Nk,p ( G ) ≤ π L (cid:0) k + ( | E | − (cid:1) . for all k ≥ in the case of the lower bound, and for all sufficiently large k in the case of the upperbound, in particular for k ≥ | E | − . In particular, (3.3) e L Nk,p ( G ) , L Nk,p ( G ) = π L k + O ( k ) as k → ∞ . This theorem follows from results in Sections 4.2 and 5.2, in particular Theorems 4.9 and 5.3(the latter in conjunction with Remark 5.4). In this case, it is possible to say a fair amount aboutwhen there is equality in the lower bound in (3.2); see Propositions 4.11 and 4.12.We can also give a description of the asymptotic behaviour of the minimal partitions realising L Dk,p , L Nk,p etc. Our main result states that asymptotically all clusters are of length of order 1 /k :no clusters can remain too “large”. SYMPTOTICS AND ESTIMATES FOR SPECTRAL MINIMAL PARTITIONS OF METRIC GRAPHS 7
Theorem 3.3.
Fix p ∈ [1 , ∞ ] and, for each k ≥ , let P Nk , e P Nk and P Dk be any admissible partitionsrealising L Nk,p ( G ) , e L Nk,p ( G ) and L Dk,p ( G ) , respectively. Denote the size of the largest cluster of eachby L N max ( k ) , e L N max ( k ) and L D max , respectively. Then (3.4) L N max ( k ) , e L N max ( k ) , L D max ( k ) = O ( k − ) as k → ∞ . Remark 3.4.
One of the main open problems in the theory of spectral minimal partitions forplanar domains Ω is the so-called hexagonal conjecture that seems to go back to Caffarelli and Lin,see [BNH17, § k →∞ L Dk,p k = λ | Ω | , where λ is the lowest eigenvalue of the Dirichlet Laplacian on a hexagon of unit area. A simplescaling argument suggests that on graphs the denominators should be replaced by their squares,whereas the correct counterpart of hexagons are intervals, since – as they get finer and finer –partitions only consist of intervals and stars, with the latter becoming irrelevant as their totalnumber is bounded by | V | . Summing up, it seems that the correct one-dimensional version of thehexagonal conjecture is lim k →∞ L Dk,p k = π L . of which Theorem 3.1 yields an analytic proof. This is complemented by Theorem 3.3, whichestablishes a certain asymptotic uniformity of the size of the intervals and stars in the optimalpartitions.Due to parallels between the respective proofs in the Dirichlet and natural cases, we will groupthe lower bounds together in Section 4 and the upper bounds in Section 5; the proof of Theorem 3.3will be given in Section 6, where we also collect a couple of results (improved bounds, Corollary 6.1,and a monotonicity statement for L Nk,p as a function of k for k sufficiently large, Theorem 6.2) whichfollow from Theorem 3.3. We also show that this monotonicity result does not necessarily hold forall k , see Example 6.3. Finally, we recall that Section 7 is devoted to the non-existence of a secondterm (i.e., term of first order) in the asymptotic expansions (3.1) and (3.3). We also set up one ofour examples to give an example that there need not be any second term in the Weyl asymptoticsfor µ k (see Remark 7.3). 4. Lower bounds
Dirichlet partitions.
We first consider lower bounds on the optimal Dirichlet partitionenergy L Dk,p ( G ). Theorem 4.1.
Let G be a compact and connected metric graph with total length L > . For any p ∈ [1 , ∞ ] and any k ≥ , we have (4.1) L Dk,p ( G ) ≥ π k L . Equality implies that G is an equilateral k -star S k . Observe that the special case of p = ∞ can also be obtained from combining [KKLM20, Prop. 5.5]and [Fri05, Thm. 1]. M. HOFMANN, J. B. KENNEDY, D. MUGNOLO, AND M. PL ¨UMER
Proof.
Since L Dk,p ( G ) is monotonically increasing in p ∈ [1 , ∞ ] (see [KKLM20, Prop. 7.1]), it sufficesto prove (4.7) for p = 1 only. We suppose that G , . . . , G k are the clusters of an optimal partitionassociated with L Dk, ( G ); then since each has at least one Dirichlet vertex, we may apply the versionof Nicaise’ inequality for Dirichlet problems cf. Proposition A.1 to obtain λ ( G i ) ≥ π / (4 |G i | ), i = 1 , . . . , k . Thus, by Jensen’s inequality in discrete form applied to the convex map x x − , x >
0, we find L Dk, ( G ) = 1 k k X i =1 λ ( G i ) ≥ π k k X i =1 |G i | − ! ≥ π k L . This proves (4.1). For the case of equality, first note that there is equality in Proposition A.1.(1)if and only if G i is an interval of length |G i | , with one Dirichlet and one Neumann endpoint(i.e., vertex); this is an immediate consequence of [Fri05, Lemma 3] together with the variationalcharacterisation of λ . Moreover, equality in Jensen’s inequality implies that |G | = . . . = |G k | = L/k . Hence equality in (4.1) (for any p ≥ k ≥
2) is only possible if all the G i areintervals of length L/k with one Dirichlet and one Neumann endpoint. Since the boundary betweenneighbouring clusters is always marked by a Dirichlet vertex, the only possible connected metricgraph that can have these graphs as partition clusters is S k . (cid:3) Remark 4.2.
The theorem contains the statement that the optimal k -partition of an equilateral k -star S k , for any p ∈ [1 , ∞ ], is the expected one, i.e., where each edge is a cluster. More interestingly,this partition reflects the nodal pattern of λ k ( S k ); and S k is also the (unique) minimiser of λ k ( G )among all graphs of fixed total length, as proved by Friedlander [Fri05]. As with Friedlander’sinequality, Theorem 4.1 implies in particular that the minimal possible values for L Dk,p ( G ) (amongall possible graphs G of given length L ) do not exhibit the asymptotic behaviour π k /L whichwould be consistent with the Weyl asymptotics of each fixed graph.In both cases, the divergence from the Weyl asymptotics is due to the factor of 1/4 appearing inNicaise’ inequality for λ , which reflects the case of the interval with only one Dirichlet endpoint. Torecover the asymptotically correct value, there needs to be a reasonable “distribution” of Dirichletvertices in the graph; in particular, an improved inequality can only be valid for sufficiently large k or for special classes of graphs. Before stating our improved estimates, we recall that a connectedmetric graph is called doubly connected if it is not simply connected as a metric space, i.e., if atleast two edges need to be deleted in order to make it disconnected. We refer to Section 7.1 for adetailed discussion of the asymptotics for equilateral stars. Definition 4.3.
Let G be a compact and connected metric graph. We will call a metric subgraph G ′ ⊂ G a doubly connected pendant of G if G ′ has non-empty interior, G ′ is doubly connected andthere is exactly one edge e ∈ E of strictly positive length connecting G ′ with its complement G \ G ′ .The set of all doubly connected pendants of G will be denoted by P . Example 4.4.
Note that Definition 4.3 explicitly requires the existence of a bridge (of positivelength) as a precondition for the existence of any doubly connected pendants. A dumbbell graph(with non-degenerate handle) has two doubly connected pendants, consisting of its two loops. Moregenerally, an m − − m -pumpkin chain (see [BKKM19, § m >
1, two doubly connectedpendants (the two m -pumpkins) but Betti number 2( m − none .Note that any two distinct doubly connected pendants are disjoint, and that necessarily theBetti number satisfies β ≥ | P | , as any cycles belonging to different doubly connected pendantsare necessarily independent. SYMPTOTICS AND ESTIMATES FOR SPECTRAL MINIMAL PARTITIONS OF METRIC GRAPHS 9
Theorem 4.5.
Let G be a compact and connected metric graph with total length L > , | N | vertices of degree one and | P | doubly connected pendants. Fix k ≥ and p ∈ [1 , ∞ ] . Then for any k ≥ | N | + | P | we have (4.2) L Dk,p ( G ) ≥ π kL (cid:0) k + 3( k − | N | − | P | ) (cid:1) . The estimate (4.2) is asymptotically sharp, in the sense that for any value of p, | N | , | P | thereexists a value of k and a family of graphs G ε such that, for these values of p, | N | , | P | , k there isequality in (4.2) as ε →
0; see Remark 4.8. The somewhat complicated case of equality in (4.2) isdiscussed in Remark 4.7.Before turning to the proof, we mention a couple of special cases. In particular, we can improvethe estimate in Theorem 4.1 by a factor 4 if G is doubly connected. Corollary 4.6.
Keep the notation and assumptions of Theorem 4.5 and denote by β := | E | − | V | +1 ≥ the Betti number of G . Then(1) for any p ∈ [1 , ∞ ] and any k ≥ with k ≥ | N | + β , (4.3) L Dk,p ( G ) ≥ π kL (cid:0) k + 3( k − | N | − β ) (cid:1) ; (2) if in addition G is itself doubly connected, then for any p ∈ [1 , ∞ ] and any k ≥ , (4.4) L Dk,p ( G ) ≥ π k L . Proof. (1) The result follows immediately from Theorem 4.5 and the previous observation that β ≥ | P | holds.(2) In this case, by assumption | N | = 0 and | P | = 0; hence (4.2) reduces to (4.4). (cid:3) Proof of Theorem 4.5.
Without loss of generality, we may assume that k > | N | + | P | , since (4.2)reduces to (4.1) for k = | N | + | P | . Firstly, as before, by monotonicity it is sufficient to show (4.2)for p = 1. So suppose that P = {G , . . . , G k } is an optimal k -partition of G for L Dk, ( G ); then atmost | N | clusters of P can contain a vertex of degree 1 and at most | P | clusters can contain adoubly connected pendant of G . Suppose j k ≤ | N | + | P | < k of the clusters admit at least one vertex of degree 1 or contain a doubly connected pendant; thenafter a renumbering if necessary we may assume that G j k +1 , . . . , G k contain neither a vertex ofdegree 1 of G nor a doubly connected pendant of G : in particular, each G i for i > j k has at leasttwo boundary vertices that are thus equipped with a Dirichlet condition, and the graph obtainedby merging all these vertices of degree 1 is doubly connected. Therefore, Proposition A.1.(2) isapplicable to these clusters, yielding λ ( G i ) ≥ π / |G i | for i > j k . Now, define(4.5) L k := j k X i =1 |G i | and note that L k < L holds, since j k < k . Then, applying Proposition A.1.(1) to the other clustersand using Jensen’s inequality as in the proof of Theorem 4.1, we see that L Dk, ( G ) = Λ D ( P ) = P j k i =1 λ ( G i ) + P ki = j k +1 λ ( G i )4 k + 3( k − j k )4 k k − j k k X i = j k +1 λ ( G i ) ≥ k k X i =1 π |G i | + 3( k − j k )4 k k − j k k X i = j k +1 π |G i | ≥ π k L + 3( k − j k )4 k π ( k − j k ) ( L − L k ) ≥ π k L + 3( k − j k )4 k π ( k − j k ) L = π kL (cid:0) k + 3( k − j k ) (cid:1) ≥ π kL (cid:0) k + 3( k − | N | − | P | ) (cid:1) . (4.6)This proves the claim. (cid:3) Remark 4.7.
Let us briefly discuss the cases of equality in (4.2). We have already seen in Theorem4.9 that equality holds for k = | N | + | P | if and only if G is the equilateral k -star. In the case k > | N | + | P | we need to analyse the estimates in (4.6). First of all, note that in this case L Dk,p ( G ) = L Dk, ( G ). Now the equalities in the fourth and sixth steps of (4.6) imply L k = 0 and | N | + | P | = j k = 0. Moreover, equality in Jensen’s inequality in the third step yields |G i | = Lk for i = 1 , . . . , k . Finally, equality in the second step, i.e., in Proposition A.1.(2), implies thatevery cluster G i of an optimal k -partition P is a caterpillar graph , i.e. a 2-regular pumpkin chainof length Lk where one of the two end points (of degree two) is equipped with Dirichlet conditions,see also Figure 4.1. Therefore, equality in (4.2) holds for k > | N | + | P | if and only if G is obtainedby arbitrarily gluing a collection of caterpillar graphs at their Dirichlet vertices so that G has novertices of degree one – in particular G has to be doubly connected. Remark 4.8.
Also note that (4.2) is asymptotically sharp if | P | > k = | P | + | N | , in thesense that there exists a family of graphs G ε differing only by their edge lengths, for which there isequality in the limit as ε →
0. To see this consider, for m ≥ n ≥
0, an equilateral m + n -stargraph where m of the degree one vertices are replaced with a loop of sufficiently small length ε >
0; when n = 0 these are the graphs considered in [KS18]. The graph W m,n thus obtained has | N | = n vertices of degree one and | P | = m doubly connected pendants. One can show that for k = m + n an optimal k -partition for L Dk,p ( W m,n ) is obtained by cutting through the centre vertex,i.e., it consists of m lasso graphs and n intervals with one Neumann and one Dirichlet vertex. Forthese graphs and k = m + n , the right-hand side of (4.2) is just π k L , corresponding to the optimalenergy L Dm + n,p of the equilateral m + n -star of total length L . If in W m,n we let the length of theloops tend to zero, then stability of λ with respect to this operation (see [BLS19]) implies that L Dk,p ( W m,n ) indeed converges to the right-hand side of (4.2).4.2. Neumann partitions.
We start with an analogue of Theorem 4.1 for Neumann partitions.In comparison with the Dirichlet case, providing a complete description of the graphs for whichthere is equality seems to be a rather difficult problem.
SYMPTOTICS AND ESTIMATES FOR SPECTRAL MINIMAL PARTITIONS OF METRIC GRAPHS 11
Figure 4.1.
A caterpillar graphwith Dirichlet vertices marked inwhite.
Figure 4.2.
The graph W m,n with m = 2 and n = 4. Theorem 4.9.
Let G be a compact and connected metric graph with total length L > . For any p ∈ [1 , ∞ ] and any k ≥ , we have (4.7) L Nk,p ( G ) ≥ e L Nk,p ( G ) ≥ π k L . If G is not a loop or if k ≥ , then there is equality if and only if there exists a rigid (respectively,a loose) k -partition whose every cluster is an interval of length L/k .Proof.
Fix k ≥
1. We give the proof for L Nk,p , since the argument for e L Nk,p is identical (note that dueto the statement about equality the statement for e L Nk,p ( G ) does not imply the full statement for L Nk,p ( G )). As in the proof of Theorem 4.1, by monotonicity in p it suffices to prove the inequality for p = 1. To this end, we suppose that G , . . . , G k are the clusters of an optimal partition associatedwith L Nk, ( G ), then(4.8) |G | + . . . + |G k | = L. Applying Proposition A.1.(1) to each cluster, we have µ ( G i ) ≥ π / |G i | for all i = 1 , . . . , k and so L Nk, ( G ) = 1 k k X i =1 µ ( G i ) ≥ π k k X i =1 |G i | ! ≥ π k k X i =1 |G i | ! − = π k L , where we have applied (4.8) and, as usual, Jensen’s inequality.Equality in (4.7) implies in particular that there is an optimising partition {G , . . . , G k } yieldingequality in the application of Proposition A.1.(1) and Jensen’s inequality. This, in turn, requiresthat the cluster G i is an interval of length L/k , for every i = 1 , . . . , k . (cid:3) Remark 4.10.
Unlike in the Dirichlet case, the condition for equality in the lower bound (4.7)does not prevent the graph from being doubly connected. In other words, we cannot expect animproved version of (4.7) for general doubly connected G . A simple example is given by the loop,for which L Nk,p ( G ) = e L Nk,p ( G ) = L Dk,p ( G ) = π k L for all k and all p .We complement Theorem 4.9 with some sufficient conditions for equality which are easy tocheck. Proposition 4.11.
Suppose that the compact and connected graph G has an Eulerian path.(1) For all p ∈ [1 , ∞ ] and all k ≥ there is equality e L Nk,p ( G ) = π k L in (4.7) .(2) If, in addition, for given k ≥ the girth s ∈ (0 , ∞ ] of G satisfies s ≥ L/k , then also L Nk,p ( G ) = π k L for all p ∈ [1 , ∞ ] . For graphs without an Eulerian path, it is still possible for there to be equality for at leastsome values of k , as the next proposition shows. It seems reasonable to expect that the equality e L Nk,p ( G ) = π k L or L Nk,p ( G ) = π k L for all k ≥ G has an Eulerian path, butwe will not explore this question here. Proof.
Suppose that G has an Eulerian path. In light of (4.7) and the monotonicity of the optimalenergies in p , it suffices to show that under the respective claimed conditions e L Nk, ∞ ( G ) , L Nk, ∞ ( G ) ≤ π k L , which we do for e L Nk, ∞ ( G ) by finding a k -partition of G having energy exactly π k /L ; we will thenshow that our chosen partition is rigid if we make the assumption about cycle lengths in G . Denoteby I the interval of length L and by I , . . . , I k the intervals obtained when I is divided into k equal subintervals of length L/k each, so that µ ( I i ) = π k /L for all i = 1 , . . . , k .Now since G contains an Eulerian path, there is a parametrisation (surjective continuous length-preserving mapping, injective apart possible from at a finite number of points corresponding tovertices of G ) φ : I → G , of such an Eulerian path. If we let P be the partition whose clusters areexactly φ ( I ) , . . . , φ ( I k ), then by construction, for any i = 1 , . . . , k e L Nk, ∞ ( G ) ≤ Λ N ∞ ( P ) = µ ( φ ( I i )) = µ ( I i ) = π k L , and ( φ ( I ) , . . . , φ ( I k )) minimises e L Nk, ∞ ( G ). If L/k ≤ s , then each cluster φ ( I ) , . . . φ ( I k ) mayself-intersect at most at its endpoint. Since k ≥ G is connected, by construction of thepartition such an endpoint is necessarily a boundary point. Hence ( φ ( I ) , . . . , φ ( I k )) is rigid and L Nk,p ( G ) = π k L . (cid:3) We finish this section with a complement to the previous proposition, which states that for everygraph G with rationally dependent edge lengths there is a sequence of values k for which there isequality e L Nk,p ( G ) = L Nk,p ( G ) = π k L . Proposition 4.12.
Assume that the edge lengths in G are pairwise rationally dependent, that is,for every pair of edges e , e ∈ E the quotient | e | / | e | is rational. Then there exists some positiveinteger m ≥ such that (4.9) e L Njm,p ( G ) = L Njm,p ( G ) = π ( jm ) L for any integer j ≥ and any p ∈ [1 , ∞ ] .Proof. As L Nk,p ( G ) ≥ e L Nk,p ( G ) both satisfy (4.7) and are monotonically decreasing in p ∈ [1 , ∞ ] forany k ≥
1, it suffices to prove existence of some integer m ≥ L Njm, ∞ ( G ) ≤ π ( jm ) L for all j ≥
1. First, we observe that the edge lengths are pairwise rationally dependent if and onlyif there is some positive real number s > m e := | e | /s is an integer for all edges e ∈ E .We set m := X e ∈ E m e = Ls .
For j ≥ P be the rigid jm -partition obtained after cutting through every vertex of G andthen dividing each edge e ∈ E into jm e intervals of equal length s/j , so P is an equipartition with L Njm, ∞ ( G ) ≤ Λ N ∞ ( P ) = π j s = π ( jm ) L . This proves the claim. (cid:3)
Remark 4.13.
In particular, the previous proposition holds for equilateral graphs, and the proofshows that in this case we may choose m as the cardinality of the edge set in that case. SYMPTOTICS AND ESTIMATES FOR SPECTRAL MINIMAL PARTITIONS OF METRIC GRAPHS 13 Upper bounds
Dirichlet partitions.
We next consider upper bounds on L Dk,p ( G ). Theorem 5.1.
Suppose G is a compact and connected metric graph. Then we have (5.1) L Dk,p ( G ) ≤ π L (cid:18) k + (cid:18) | E | − − (cid:22) | N | (cid:23)(cid:19)(cid:19) for all sufficiently large integers k ≥ and all p ∈ [1 , ∞ ] , where | N | denotes the number verticesin G of degree . In particular, (5.1) holds whenever k ≥ Lℓ min + | E | − , where we recall that ℓ min = min e ∈ E | e | is the minimal edge length.Proof. By monotonicity, it suffices to prove the theorem for p = ∞ . The proof consists of construct-ing a “test partition” formed by dividing each edge into a given number of intervals in accordancewith its length, where the lengths are suitably chosen.Without loss of generality, we may assume that G has at least two edges, otherwise G wouldbe a cycle or an interval and in both cases (5.1) is obviously satisfied. Let E N denote the set ofpendant edges in E , i.e those edges containing a vertex of degree one. Note that, since G has atleast two edges and G is connected, each edge contains at most one vertex of degree one, and thus | E N | = | N | holds. Fix an integer n ≥ Ln ≤ | e | for all e ∈ E . Now for each e ∈ E there exists an integer m e such that(5.2) m e · Ln ≤ | e | < ( m e + 1) Ln , if e ∈ E \ E N and(5.3) 2 m e − · Ln ≤ | e | < m e + 12 · Ln if e ∈ E N . For e ∈ E \ E N we then partition e into m e intervals of equal length | e | m e , and for e ∈ E N we partition e into m e intervals, so that the interval containing the vertex of degree one has length | e | m e +1 and the remaining intervals have length | e | m e +1 . Note that the interval lengths here are chosenso that the first Dirichlet eigenvalue of the longer intervals and the first mixed Dirichlet–Neumanneigenvalue of the shorter intervals are both equal to π (2 m e +1) | e | . Let P be the m -partition thusobtained, where m := X e ∈ E m e . Summing up (5.2) and (5.3) and using m = P e ∈ E m e and L = P e ∈ E | e | , we immediately obtain(5.4) m − j | N | k ≤ n ≤ m + | E | − − j | N | k . By choice of the interval lengths we haveΛ D ∞ ( P ) ≤ max (cid:18) max ≤ j ≤| N | π (2 m j + 1) L j , max | N | +1 ≤ j ≤| E | π m j L j (cid:19) ≤ π n L , and thus L Dm, ∞ ( G ) ≤ π n L . Since m ≥ n − | E | + 1 + (cid:4) | N | (cid:5) and L Dk, ∞ ( G ) is monotonically increasingin k by [KKLM20, Remark 4.11], we thus have L Dn −| E | +1+ ⌊ | N | ⌋ , ∞ ( G ) ≤ π n L . Setting k := n + | E | − − ⌊ | N | ⌋ in the above inequality yields (5.1). (cid:3) Remark 5.2.
It is known that L Dk,p ( G ) dominates the k -th lowest eigenvalue µ k of the Lapacianwith natural vertex conditions, cf. [KKLM20, Prop. 5.5]. Hence, in particular, Theorem 5.1 yields,for sufficiently large k , µ k ≤ π L (cid:18) k − | E | − (cid:22) | N | (cid:23)(cid:19) . This estimate can be compared with the upper bound obtained in [BKKM17, Thm. 4.9], which inthe present case of Laplacians with no Dirichlet boundary conditions reads µ k ≤ π L (cid:18) k −
12 + 32 | E | − | V | + | N | (cid:19) ;studying the class of graphs W m,n (see Example 4.4), the latter bound was shown to be asymp-totically sharp in [KS18, Theorem 2].5.2. Neumann partitions.
Our main upper bound in this case reads as follows.
Theorem 5.3.
Suppose there exists an n -partition of G such that every associated cluster G j hasan Eulerian path, then we have (5.5) e L Nk,p ( G ) ≤ L Nk,p ( G ) ≤ π L (cid:0) k + ( n − (cid:1) for all sufficiently large integers k ≥ and all p ∈ [1 , ∞ ] . Concretely, we may take k ≥ max { | E | + n − , L } , where | E | is the number of edges of G and s ∈ (0 , ∞ ] its girth. Remark 5.4.
Obviously we may always choose n to be the number of edges of G in the previoustheorem, leading to the bound e L Nk,p ( G ) ≤ L Nk,p ( G ) ≤ π L (cid:0) k + ( | E | − (cid:1) . This is valid for all k ≥ | E | −
1, as an inspection of the proof shows that s may be replaced bythe quantity max s ( G j ), where s ( G j ) is the girth of G j , which in the case of each G j being an edgeis simply ∞ . (We still expect this bound on k , like the one in Theorem 5.3, to be far from optimalin general.) Lemma 5.5.
Given an n -partition of G with associated clusters G , . . . , G n we have (5.6) L Nm,p ( G ) ≤ n X j =1 m j m L Nm j ,p ( G j ) p ! /p if ≤ p < ∞ , max j =1 ,...,k L Nm j , ∞ ( G j ) if p = ∞ for integers m j ≥ and m = P nj =1 m j . An analogous statement holds for e L Nm,p ( G ) . SYMPTOTICS AND ESTIMATES FOR SPECTRAL MINIMAL PARTITIONS OF METRIC GRAPHS 15
Proof.
We restrict ourselves to the case 1 ≤ p < ∞ and rigid partitions, since the other cases canbe dealt with analogously. For each j we choose an optimal rigid m j -partition P j of G j associatedwith L Nm j ,p ( G j ) with clusters G ij for i = 1 , . . . , m j . We consider the induced rigid m -partition P of G given by P := n [ j =1 P j . By optimality of P j we have m j L Nm j ,p ( G j ) p = m j X i =1 µ ( G ij ) p . Thus, we obtain L Nm,p ( G ) ≤ Λ Np ( P ) = m n X j =1 m j X i =1 µ ( G ij ) p ! /p = n X j =1 m j m L Nm j ,p ( G j ) p ! /p . This concludes the proof. (cid:3)
Proof of Theorem 5.3.
Again, we may restrict ourselves to L Nk,p ( G ) and the case p = ∞ . Similarlyto the proof of Theorem 5.1, we construct a test partition dividing each Eulerian path into intervalsof equal length. Let k ≥ n be an arbitrary, sufficiently large integer with Lk ≤ |G j | for j = 1 , . . . , n .For j = 1 , . . . , n there exists an integer m j ≥
2, so that(5.7) m j · Lk ≤ |G j | < ( m j + 1) Lk .
We set m := P nj =1 m j . As in the proof of Theorem 5.1, it is immediate that(5.8) m ≤ k ≤ m + n − . Since G j has an Eulerian path and every cycle in G j has length at least s ≥ L k ≥ m j + 1 m j · Lk ≥ |G j | m j (if it has any cycles at all), we may apply the result of Proposition 4.11 to obtain L Nm j , ∞ ( G j ) = π m j |G j | . Thus, Lemma 5.5, the previous equality and Lemma 5.5 yield L Nm, ∞ ( G ) ≤ max j =1 ,...,k L Nm j , ∞ ( G j ) = max j =1 ,...,k π m j |G j | ≤ π k L . Since L Nm, ∞ ( G ) is monotonically increasing in m for sufficiently large m , in particular for m ≥ | E | (see [KKLM20, Proposition 4.14] and its proof, and note that under the assumption k ≥ | E | + n − m ≥ | E | ), we may use (5.8) to conclude L Nk − n +1 , ∞ ( G ) ≤ L Nm, ∞ ( G ) ≤ π k L . Finally, replacing k by k + n − L Nk, ∞ ( G ) ≤ π ( k + n − L = π k L + 2 π ( n − kL + π ( n − L . This concludes the proof. (cid:3) Asymptotic behaviour of the optimal partitions
In this section we give the proof of Theorem 3.3, which establishes that the maximal clustersize of any optimal partition tends to zero as k → ∞ ; this relies on the asymptotic behaviour ofthe optimal energies obtained in the previous sections. We will also give a couple of consequencesof this result, as it in turn allows us to refine and sharpen certain statements from the previoussections. Proof of Theorem 3.3.
We first give the proof in the Dirichlet case. Notationally, for any k ≥ p ∈ [1 , ∞ ] we suppose P ∗ k,p = {G , . . . , G k } to be any admissible k -partition realising L Dk,p ( G ).Fix p ∈ [1 , ∞ ]. As noted in the proof of Theorem 4.5, there are at most | N | + | P | clusters of P ∗ k,p which contain either a vertex of degree 1 or a doubly connected pendant of G . Denote by j k ≤ | N | + | P | + 1 the number of such clusters of P ∗ k,p , plus any cluster of maximal size if thereis not already at least one such cluster among them, and suppose without loss of generality thatthese clusters are numbered 1 , . . . , j k . Finally, denote by L k the total length of these j k clusters;then by construction L D max ( k ) ≤ L k . We will prove that in fact L k = O ( k − ) as k → ∞ .Firstly, observe that(6.1) Λ D ( P ∗ k,p ) = π L k + O ( k ) as k → ∞ , since by monotonicity in p L Dk,p ( G ) = Λ Dp ( P ∗ k,p ) ≥ Λ D ( P ∗ k,p ) ≥ L Dk, ( G )and both L Dk,p ( G ) and L Dk, ( G ) behave like π L k + O ( k ) as k → ∞ , by Theorem 3.1. Now, withthe notation described above, for k > j k , using that λ ( G i ) ≥ π |G i | for all i = 1 , . . . , j k and λ ( G i ) ≥ π |G i | for all i = j k + 1 , . . . , k , the usual argument (see (4.6)) yieldsΛ D ( P ∗ k,p ) ≥ π π k L + 3 π k − j k ) k ( L − L k ) for all k > j k . Suppose now that L k = O ( k − ), so that, possibly up to a subsequence, lim k →∞ kL k = ∞ . We consider the asymptotic behaviour of this subsequence of k ; our goal is to show that in theasymptotic limit this expression must be larger than allowed by (6.1). Since j k remains bounded,the first term in the above estimate converges to zero, and so is certainly of order O (1), while( k − j k ) k ( L − L k ) = k ( L − L k ) + O ( k ) as k → ∞ . But since k ( L − L k ) = k L − L k L ) = k L (cid:18) L L k + O ( L k ) (cid:19) as k → ∞ and lim k →∞ kL k = ∞ by assumption, this means thatΛ D ( P ∗ k,p ) = π L k + O ( k ) as k → ∞ , a contradiction to (6.1).In the natural cases, the argument is similar but simpler owing to the better estimate µ ( G i ) ≥ π |G i | for all i . We consider L k := L N max ( k ); the case e L N max ( k ) is identical. We fix p ∈ [1 , ∞ ] and take P ∗ k,p = {G , . . . , G k } to be an optimal k -partition realising L Nk,p ( G ) and suppose that the cluster G SYMPTOTICS AND ESTIMATES FOR SPECTRAL MINIMAL PARTITIONS OF METRIC GRAPHS 17 has size |G | = L N max ( k ). As in the Dirichlet case, due to the asymptotics (3.3) of Theorem 3.2 wehave(6.2) Λ N ( P ∗ k,p ) = π L k + O ( k ) as k → ∞ . On the other hand, for k ≥ N ( P ∗ k,p ) ≥ π k |G | + k − k k − k X i =2 |G i | − !! ≥ π kL k + π ( k − k ( L − L k ) . Under the assumption that L k = O ( k − ), the same argument as in the Dirichlet case now yieldsthat, possibly up to a subsequence, Λ N ( P ∗ k,p ) = π L k + O ( k ) as k → ∞ , contradicting (6.2). (cid:3) As a first corollary of Theorem 3.3 we obtain an improved version of the lower bound in Theo-rem 3.1 for sufficiently large k ; namely, we can drop the term β appearing there. Corollary 6.1.
Let G be a compact and connected metric graph with total length L > and | N | vertices of degree one. Fix p ∈ [1 , ∞ ] . Then there exists k ≥ such that for all k ≥ k we have (6.3) L Dk,p ( G ) ≥ π kL (cid:0) k + 3( k − | N | ) (cid:1) . Proof.
By monotonicity it is sufficient to prove the assertion for p = 1. For k ≥
2, we supposethat P Dk is an admissible k -partition realising L Dk, ( G ) and L D max ( k ) is the maximum length of theclusters in P Dk . By Theorem 3.4 we find some k ≥ L D max ( k ) < ℓ min holds for all k ≥ k . In particular, the clusters appearing in P Dk are either intervals or stars,where all non-centre vertices are cut points. Let G , . . . , G | N | be the clusters of P Dk that containthe vertices of G of degree one and let G | N | +1 , . . . , G k be the remaining clusters. We then have λ ( G j ) = π |G j | for j = 1 , . . . , | N | and λ ( G j ) ≥ π |G j | for j = | N | + 1 , . . . , k by (A.2). Adapting thearguments in (4.6) we obtain L Dk, ( G ) = Λ D ( P Dk ) ≥ π kL (cid:0) k + 3( k − | N | ) (cid:1) . (cid:3) As a second consequence of Theorem 3.3 we will prove that, for fixed p ∈ [1 , ∞ ], L Nk,p is amonotonically increasing function of k , at least for k sufficiently large. Theorem 6.2.
Let G be a compact and connected graph, and fix p ∈ [1 , ∞ ] . Then there exists k ≥ depending only on G and p such that L Nk ,p ( G ) ≥ L Nk ,p ( G ) for all k ≥ k ≥ k . We recall that the monotonicity in the loose case, e L Nk ,p ( G ) ≥ e L Nk ,p ( G ) for all k ≥ k ≥
1, wasalready established in Remark 4.11 of [KKLM20], as was Theorem 6.2 in the special case p = ∞ in [KKLM20, Proposition 4.14] (which was also required in one of the above proofs). In generalwe cannot necessarily expect k = 1, see Example 6.3. Proof.
Since the case p = ∞ was treated in [KKLM20], we give the proof for p ∈ [1 , ∞ ). So fix p ∈ [1 , ∞ ) and for k ≥ P ∗ k,p = {G , . . . , G k } any rigid k -partition achieving L Nk,p ( G ). ByTheorem 3.3 there exists some k = k ( G , p ) such that for every k ≥ k every cluster of P ∗ k,p haslength strictly shorter than the shortest edge length of G , and in particular every cluster is a tree,which meets any neighbouring cluster of P ∗ k,p at a single vertex.It clearly suffices to prove the theorem for k = k + 1. Fix k ≥ k + 1 and consider P ∗ k,p ; wesuppose without loss of generality that(6.4) µ ( G k ) = max i =1 ,...,k µ ( G i )and that G k − is a neighbour of G k . We now set e G k − := G k − ∪ G k ; then since G k − and G k necessarily meet at a single point, by [BKKM19, Theorem 3.10(1)], we have µ ( e G k − ) ≤ µ ( G k − ).We construct a test k − e P := {G , . . . , G k − , e G k − } of G ; then, again using the fact that G k − and G k meet at a single point and P ∗ k,p was assumed rigid, e P is a rigid k − G .We claim that Λ Np ( P ∗ k,p ) ≥ Λ Np ( e P ), from which the conclusion of the theorem in the case p ∈ [1 , ∞ ) will immediately follow. In fact, this is an elementary calculation using (6.4): it followsfrom (6.4) that µ ( G k ) p ≥ k − k − X i =1 µ ( G i ) p , and henceΛ Np ( P ∗ k,p ) p − Λ Np ( e P ) p = 1 k k X i =1 µ ( G i ) p − k − k − X i =1 µ ( G i ) p + µ ( e G k − ) p ! = 1 k µ ( G k ) p − k ( k − k − X i =1 µ ( G i ) p + 1 k µ ( G k − ) p − k − µ ( e G k − ) p ≥ k µ ( G k ) p − k ( k − k − X i =1 µ ( G i ) p since µ ( G k − ) ≥ µ ( e G k − ). By (6.4), this latter expression is nonnegative, and so we conclude thatΛ Np ( P ∗ k,p ) ≥ Λ Np ( e P ), as desired. (cid:3) Example 6.3.
We consider the graph G of [KKLM20, Example 8.2], depicted in Figure 6.1; weclaim that for this graph L N ,p ( G ) < L N ,p ( G ) for all p ∈ [1 , ∞ ], that is, monotonicity in Theorem 6.2fails when k = 1 and k = 2. Figure 6.1.
The graph G for which L N ,p ( G ) < L N ,p ( G ) for all p ∈ [1 , ∞ ].Suppose that G has total length L and fix p ∈ [1 , ∞ ]. It was already shown in [KKLM20,Example 8.2] that L N ,p ( G ) = π L . Next, we note that by definition L N ,p ( G ) = µ ( G ). Now by SYMPTOTICS AND ESTIMATES FOR SPECTRAL MINIMAL PARTITIONS OF METRIC GRAPHS 19 the Band–L´evy inequality, Proposition A.1(2), since G is not a 2-regular pumpkin chain, we have µ ( G ) > π L . This proves the claimed reverse monotonicity.7. Asymptotics on two simple graphs
In the previous sections, we proved that the minimal energies L N,Dk,p ( G ) satisfy the Weyl-typeasymptotic law L N,Dk,p ( G ) = π L k + O ( k ) as k → ∞ . In this section we are going to discuss the behaviour of the first order term O ( k ) in this expansion.A natural question to ask is if there exists some c ∈ R such that L N,Dk,p ( G ) = π L k + ck + O (1) as k → ∞ holds. We are going to show that in general such c does not exist. More precisely, we study thesequence given by c k := L N,Dk,p ( G ) − π k L k , k ∈ N and give examples where ( c k ) k has a limit points for some given a ∈ N (equilateral star graphs with2 a edges) or uncountably many limit points (two disjoint path graphs with rationally independentlengths). For simplicity of our discussion, we restrict ourselves to the case p = ∞ , but note thatour techniques may easily be adapted to the case p ∈ [1 , ∞ ).7.1. Equilateral stars.
For m ≥
3, we consider the equilateral m -star S m of total length L . Lemma 7.1.
For j ∈ N we have L Djm +1 , ∞ ( S m ) = µ jm +1 ( S m ) = π m j L , L Djm + r, ∞ ( S m ) = µ jm + r ( S m ) = π m ( j + ) L , r = 2 , . . . , m. Proof.
The ordered eigenvalues µ k ( S m ) of the equilateral m -star S m are µ jm +1 ( S m ) = π m j L µ jm + r ( S m ) = π m ( j + ) L , r = 2 , . . . , m (7.1)for j ∈ N (cf. [Fri05, Example 3]). By [KKLM20, Proposition 5.5] we have µ k ( S m ) ≤ L Dk, ∞ ( S m ) for k ∈ N . Therefore it will be sufficient to find respective partitions of S m whose energies coincideswith the eigenvalues in (7.1) and these partitions will be optimal.For k = jm + 1 we consider the partition P consisting of an equilateral m -star with edge length L mj , m intervals of length L mj each having one Dirichlet and one Neumann vertex and m ( j − Lmj each having two Dirichlet vertices. Then each cluster of P has the sameDirichlet energy π m j L and we conclude L Dk, ∞ ( S m ) = Λ Dp ( P ) = π m j L . For k = mj + r with 1 < r ≤ m we consider a partition P obtained after cutting through thecenter vertex of the star, where the first r edges e , . . . , e r are divided into j + 1 intervals – one oflength Lm (2 j +1) with one Neumann and one Dirichlet vertex and the other j of length Lm (2 j +1) withtwo Dirichlet vertices – and the remaining m − r edges e r +1 , . . . , e m are divided into j intervals –one of length Lm (2 j − with one Neumann and one Dirichlet vertex and the other j of length Lm (2 j − with two Dirichlet vertices. The Dirichlet energy of the clusters in e , . . . , e r is π m ( j + ) L whereasthe Dirichlet energy of the clusters in e r +1 , . . . , e m is π m ( j − ) L . We obtain L Dk, ∞ ( S m ) = Λ D ∞ ( P ) = π m ( j + ) L . This concludes the proof. (cid:3)
Figure 7.1.
The optimal 7-, 8- and 9-partitions of the 3-star in the proof ofLemma 7.1. White vertices denote vertices with Dirichlet conditions.
Proposition 7.2.
The limit set of the sequence ( c k ) k ∈ N with c k := L Dk, ∞ ( S m ) − π k L k , k ∈ N , is (7.2) (cid:26) − π L (cid:27) ∪ (cid:26) π ( s − − m ) L (cid:12)(cid:12) s = 1 , . . . , m − (cid:27) . In particular, ( c k ) k ∈ N has m − limit points if m is even and m limit points if m is odd.Proof. The assertion immediately follows from Lemma 7.1 if one considers the subsequences( c k j ) j ∈ N given by k j := jm + r for r = 1 , . . . , m and j ∈ N . Indeed, for r = 1, we have k j c k j = π m j L − π k j L = π L (cid:2) ( k j − − k j (cid:3) = π L ( − k j + 1)and, thus, c k j → − π L as k j → ∞ . For 1 < r ≤ m , we have k j c k j = π m ( j + ) L − π k j L = π L (cid:20)(cid:16) k j + m − r (cid:17) − k j (cid:21) = π L (cid:20) k j (cid:16) m − r (cid:17) + (cid:16) m − r (cid:17) (cid:21) and, thus, c k j → π ( m − r ) L as k j → ∞ . Note that, if m is even, the limit point in the second casecoincides with the one in the first case for r = m + 1. (cid:3) SYMPTOTICS AND ESTIMATES FOR SPECTRAL MINIMAL PARTITIONS OF METRIC GRAPHS 21
Remark 7.3.
Proposition 7.2 also shows that, if we write µ k ( S m ) = π k L + c k k, then the set of points of accumulation of ( c k ) k ∈ N is exactly (7.2). This is an immediate consequenceof the equality L Dk, ∞ ( S m ) = µ k ( S m ) for all k ≥
1, as shown in Lemma 7.1. In particular, we havean explicit example for the non-existence of a second term in the Weyl asymptotics for µ k .We now consider the case of natural partitions. Lemma 7.4.
For j ∈ N we have L Njm + r, ∞ ( S m ) = π m ( j + ) L , r = 1 , . . . , (cid:4) m (cid:5) ,π m ( j + 1) L , r = (cid:4) m (cid:5) + 1 , . . . , m. (7.3) Proof.
We set k = jm + r . We first show that Λ N ∞ is indeed bounded from below by the termsappearing on the right-hand-side of (7.3) respectively. For an arbitrary k -partition P of S m , let P ′ denote the set of clusters in P that intersect at least two edges of S m and, for each edge e i of S m , let P i denote the set of clusters in P that only intersect e i . Furthermore, let k ′ = |P ′ | and k i := |P i | . By choice of k ′ and k i , we have k = k ′ + P mi =1 k i and k ′ ≤ m , where the latter holds,since each edge of S m intersects at most one of the clusters in P ′ . All clusters in P i are intervals,so we may assume that each element of P i has the same length ℓ i . (Note that we only decreaseΛ N ∞ if we adjust the length of the single intervals, so that all off them have the same length.) Inparticular, we have µ ( G i ) = π ℓ i for all G i ∈ P i .Now, let us first consider the case 1 ≤ r ≤ m . Without loss of generality, we may assume that ℓ i > Lm ( j + ) holds for i = 1 , . . . , m – otherwise, Λ N ∞ ( P ) ≥ π m ( j + ) L would obviously be satisfied.We obtain Lm ≥ X G i ∈P i |G i | = k i ℓ i > k i Lm ( j + )and, thus, k i ≤ j for i = 1 , . . . m . This, in turn, implies k ′ = k − m X i =1 k i ≥ jm + r − jm = r ≥ , i.e. P ′ is non-empty. We consider an arbitrary element G ′ ∈ P ′ . For i = 1 , . . . , m with | e i ∩ G ′ | > | e i ∩ G ′ | = Lm − k i ℓ i < Lm − jLm ( j + ) = L m ( j + ) . Thus, G ′ is a metric star whose maximum length ℓ max ( G ′ ) is bounded from above by L m ( j + ) . Weobtain Λ N ∞ ( P ) ≥ µ ( G ′ ) ≥ π ℓ max ( G ′ ) > π m ( j + ) L , where the second step follows from [ACS18, Lemma 3.3].Next, we consider the case m < r ≤ m . First note that Λ N ∞ ( P ) ≥ π m ( j +1) L is obviously satisfiedif ℓ i ≤ Lm ( j +1) holds. On the other hand, the case ℓ i > Lm ( j +1) for all i does not occure, since thenfollowing the argumentation of the first case yields k ′ ≥ r > m , which is a contradiction to k ′ ≤ m ,as we stated in the the beginning of the proof. Altogether, we have seen that L Nk, ∞ ( S m ) is indeed bounded from below by the terms appearingon the right-hand-side. To show equality, we simply present k -partitions with Neumann energyequal to the right-hand-side – obviously, these partitions are spectral minimal partitions. In thecase 1 ≤ r < m , we make a choice of r pairs of edges and consider their respective unions e ∪ e , . . . , e r − ∪ e r ; each of these unions is an Eulerian path in S m . Now let P be the partitionwhere each of these unions is decomposed into 2 j + 1 intervals of equal length Lm ( j + ) and everyother edge e i , i > r is decomposed into j intervals of length Lmj (see the decomposition on the leftin Figure 7.2). This partition has Neumann energy Λ N ∞ ( P ) = π m ( j + ) L . In the case m < r ≤ m ,we consider the jm + r -partition that decomposes the first r edges into j + 1 intervals of length Lm ( j +1) and the latter m − r edges into j intervals of length Lmj (see the two decompositions on theright in Figure 7.2). Again, this partition has the desired Neumann energy. (cid:3)
Figure 7.2.
The optimal 7-, 8- and 9-partitions of the 3-star in the proof of Lemma 7.4.
Remark 7.5.
Note that the spectral minimal partitions in the proof of Lemma 7.4 are not unique.For example, another optimal jm + 1-partition – whose topology differs from the one presentedin the proof – is obtained by decomposing S m into one equilateral m -star of total length L j +1 and jm intervals of length Lm ( j + ) (see Figure 7.3). In fact, this choice seems to be more natural, sinceeach cluster has the same Neumann energy. Figure 7.3.
A different optimal 7-partitions of the 3-star.
SYMPTOTICS AND ESTIMATES FOR SPECTRAL MINIMAL PARTITIONS OF METRIC GRAPHS 23
Remark 7.6.
The m -star S m can be covered with m Eulerian paths, if m is even, and m +12 Eulerianpaths, if m is odd. Therefore, Theorem 5.3 yields the upper bounds L Nk, ∞ ( S m ) ≤ π ( k + m − L , if m is even, π ( k + m +12 − L , if m is odd.Lemma 7.4 shows that these bounds are actually sharp if m is even and k = mj + 1, or m is oddand k = mj + m +12 for j ∈ N respectively. Proposition 7.7.
The limit set of the sequence ( c k ) k ∈ N with c k := L Nk, ∞ ( S m ) − π k L k , k ∈ N , is { } ∪ (cid:26) π sL (cid:12)(cid:12) s = 1 , . . . , m (cid:27) , if m is even, and { } ∪ (cid:26) π sL (cid:12)(cid:12) s = 1 , . . . , m − (cid:27) ∪ (cid:26) π ( t − ) L (cid:12)(cid:12) t = 1 , . . . , m − (cid:27) if m is odd. In particular, ( c k ) k ∈ N has m limit points if m is even and m limit points if m is odd.Proof. This immediately follows from Lemma 7.1 if one considers the subsequences ( c k j ) j ∈ N givenby k j := jm + r for r = 1 , . . . , m and j ∈ N . Indeed, calculations entirely analogous to the onesin the proof of Proposition 7.2 show that c k j → π ( m − r ) L as k j → ∞ for r = 1 , . . . , (cid:4) m (cid:5) , while c k j → π (2 m − r ) L for r = (cid:4) m (cid:5) + 1 , . . . , m . Finally, we remark that if m is even, then the limit pointsin the two cases coincide (replace r with r + m ), whereas they are distinct if m is odd. (cid:3) Two disjoint intervals with rationally independent lengths.
Let G a = I ⊔ I a be thedisjoint union of the intervals I := [0 , , I a := [0 , a ]for some a >
0. Recall that(7.4) π ( a + 1) k ≤ L Nk, ∞ ( G a ) ≤ π ( a + 1) k + 2 π ( a + 1) k + π ( a + 1) holds for k ≥ c k ) k ≥ given by(7.5) c k = L Nk, ∞ ( G a ) − π k ( a +1) k , k ≥ . First note that we have(7.6) 0 ≤ c k ≤ π ( a + 1) for k ≥ c k ) k ≥ is the whole interval [0 , π ( a +1) ] if a is irrational. In order to show this, let us first compute the minimal energy L Nk, ∞ ( G a ) for k ≥
2. Of course, for given i ∈ { , . . . , k − } , an optimal k -partition of the form P = ( G , . . . , G i , G i +1 , . . . , G k )for Λ N ∞ with G , . . . , G i ⊂ I , G i +1 , . . . , G k ⊂ I a . is obtained by taking each cluster in I of equal length i and each cluster in I a of equal length ak − i , that is,(7.7) L Nk, ∞ ( G a ) = min ≤ i ≤ k − max (cid:26) π i , π ( k − i ) a (cid:27) . Let us further investigate (7.7). One easily sees that(7.8) max (cid:26) π i , π ( k − i ) a (cid:27) = π ( k − i ) a , i ≤ (cid:22) ka + 1 (cid:23) π i , i ≥ (cid:24) ka + 1 (cid:25) . In particular, we have L Nk, ∞ ( G a ) = min ≤ i ≤ k − max (cid:26) π i , π ( k − a (cid:27) = min ( min ≤ i ≤⌊ ka +1 ⌋ π ( k − i ) a , min ⌈ ka +1 ⌉≤ i ≤ k − π i ) = min ( π ⌈ aa +1 k ⌉ a , π (cid:18)(cid:24) ka + 1 (cid:25)(cid:19) ) . (7.9)We can treat the asymptotics via study of the orbit of the rotation map T α : R / Z → [0 , T α x = x + α mod 1 . It is a well-known fact that the orbits of the map T α are dense in [0 ,
1] if and only if α ∈ R \ Q (see[Dev89, Theorem 3.13]). Theorem 7.8.
Let c k , k ≥ , be defined as in (7.5) . If a ∈ Q , then ( c k ) k ≥ has a finite limit set;if a ∈ R \ Q , then the limit set of ( c k ) k ≥ is the whole interval [0 , π ( a +1) ] .Proof. Due to (7.9), we have(7.11) c k = min π (cid:16)(cid:6) ka +1 (cid:7) − k ( a +1) (cid:17) k , π (cid:16) a (cid:6) aka +1 (cid:7) − k ( a +1) (cid:17) k . We compute π (cid:16)(cid:6) ka +1 (cid:7) − k ( a +1) (cid:17) k = π k (cid:18)(cid:24) ka + 1 (cid:25) − ka + 1 (cid:19) (cid:18)(cid:24) ka + 1 (cid:25) + ka + 1 (cid:19) = (cid:18)(cid:24) ka + 1 (cid:25) − ka + 1 (cid:19) (cid:18) π a + 1 + π k (cid:18)(cid:24) ka + 1 (cid:25) − ka + 1 (cid:19)(cid:19) = T k aa +1 (0) (cid:18) π a + 1 + π k T k aa +1 (0) (cid:19) = 2 π a + 1 T k aa +1 (0) + o (1) as k → ∞ (7.12) SYMPTOTICS AND ESTIMATES FOR SPECTRAL MINIMAL PARTITIONS OF METRIC GRAPHS 25 and π (cid:16) a (cid:6) aka +1 (cid:7) − k ( a +1) (cid:17) k = π a k (cid:18)(cid:24) aka + 1 (cid:25) − aka + 1 (cid:19) (cid:18)(cid:24) aka + 1 (cid:25) + aka + 1 (cid:19) = (cid:18)(cid:24) aka + 1 (cid:25) − aka + 1 (cid:19) (cid:18) π a ( a + 1) + π a k (cid:18)(cid:24) ka + 1 (cid:25) − ka + 1 (cid:19)(cid:19) = T k a +1 (0) (cid:18) π a ( a + 1) + π a k T k a +1 (0) (cid:19) = 2 π a ( a + 1) T k a +1 (0) + o (1) as k → ∞ . (7.13)Since the orbits of T a +1 and T aa +1 are periodic if and only if a ∈ Q , we deduce that a has a finitelimit set if and only if a ∈ Q . Suppose a ∈ R \ Q , then T k a +1 (0) + T k aa +1 (0) = ka + 1 − (cid:22) ka + 1 (cid:23) + aka + 1 − (cid:22) aka + 1 (cid:23) = k − (cid:22) ka + 1 (cid:23) − (cid:22) aka + 1 (cid:23) = (cid:24) aka + 1 (cid:25) − (cid:22) aka + 1 (cid:23) = 1 . (7.14)Let x ∈ [0 , k n ) n ∈ N is a strictly increasing sequence with(7.15) lim n →∞ T k n a +1 (0) = x, then with (7.11), (7.12) and (7.13) for all k ∈ N we inferlim n →∞ c k n = min (cid:26) π (1 − x ) a + 1 , π xa ( a + 1) (cid:27) = π xa ( a + 1) , x ≤ aa + 12 π (1 − x ) a + 1 , x > aa + 1 , (7.16)and hence the limit set of ( c k ) k ≥ is dense in [0 , π ( a +1) ]. Since the limit set is clearly closed, weconclude that it equals [0 , π ( a +1) ]. (cid:3) In the Dirichlet case, we may similarly consider the limit set of the sequence ( c k ) k ≥ given by(7.17) c k = L Dk, ∞ ( G a ) − π k (1+ a ) k , k ≥ . On an interval I = [0 , ℓ ] we have(7.18) L Dk +1 , ∞ ( I ) = L Nk, ∞ ( I ) , k ≥ , which directly gives us the following result. Theorem 7.9.
Let c k , k ≥ , be defined as in (7.17) . If a ∈ Q , then ( c k ) k ≥ has a finite limit set;if a ∈ R \ Q , then the limit set of ( c k ) k ≥ is the interval [ − π ( a +1) , − π ( a +1) ] .Proof. Using (7.18) yields(7.19) L Dk +2 , ∞ ( G a ) = L Nk, ∞ ( G a ) and, thus, L Dk +2 , ∞ ( G a ) − π ( k +2) (1+ a ) k + 2 = kk + 2 L Nk, ∞ ( G a ) − π k ( a +1) k − π ( a + 1) + o (1) as k → ∞ . The assertion now follows immediately from Theorem 7.8. (cid:3)
Appendix A. Isoperimetric inequalities
The first isoperimetric inequality for metric graphs was discovered by Nicaise 35 years ago; it issharp, as shown by Friedlander 20 years later. However, it has been observed by several authorsthat special classes of graphs allow for improved isometric inequalities: to help keep the paper moreself-contained and since we make use of some of them repeatedly, we list here the most relevant.
Proposition A.1.
Let G be any compact connected metric graph. Then the following assertionshold.(1) We have (A.1) λ ( G ) ≥ π |G| and µ ( G ) ≥ π |G| , where in the first case G is equipped with at least one Dirichlet vertex. Equality in eitherinequality implies that G is a path graph (interval) of length |G| , with a Dirichlet vertex atexactly one endpoint and a natural (Neumann) condition at the other in the first case, andnatural conditions at both endpoints in the second case.(2) If additionally (possibly upon identifying all Dirichlet vertices) G is doubly connected, then wehave (A.2) λ ( G ) ≥ π |G| and µ ( G ) ≥ π |G| , In this case, equality is attained only by 2-regular pumpkin chains (second case), or 2-regularpumpkin chains with two edges of equal length attached to one of the endpoints and the degen-erate case of an interval with two Dirichlet endpoints ( caterpillar graphs , first case).
The inequalities in (1) may be found in [Nic87, Th´eor`eme 3.1]. For the characterisation of equal-ity, see for example [Fri05, Theorem 1]. For the inequalities in (2) we refer to [BL17, Theorem 2.1]and [BKKM17, Theorem 3.4 and Lemma 4.3].
References [ACS18] O. Amini and D. Cohen-Steiner,
A transfer principle and applications to eigenvalue estimates for graphs ,Comment. Math. Helv. (2018), 203–223.[BBRS12] R. Band, G. Berkolaiko, H. Raz, and U. Smilansky, The number of nodal domains on quantum graphsas a stability index of graph partitions , Comm. Math. Phys. (2012), 815–838.[BL17] R. Band and G. L´evy,
Quantum graphs which optimize the spectral gap , Ann. Henri Poincar´e (2017),3269–3323.[BKKM17] G. Berkolaiko, J. B. Kennedy, P. Kurasov, and D. Mugnolo, Edge connectivity and the spectral gap ofcombinatorial and quantum graphs , J. Phys. A: Math. Theor. (2017), 365201.[BKKM19] G. Berkolaiko, J. B. Kennedy, P. Kurasov and D. Mugnolo, Surgery principles for the spectral analysisof quantum graphs , Trans. Amer. Math. Soc. (2019), 5153–5197.[BLS19] G. Berkolaiko, Yu. Latushkin and S. Sukhtaiev,
Limits of quantum graph operators with shrinking edges ,Adv. Math. (2019), 632–669.[BE09] J. Bolte and S. Endres,
The trace formula for quantum graphs with general self-adjoint boundary con-ditions , Ann. Henri Poincar´e (2009), 189–223. SYMPTOTICS AND ESTIMATES FOR SPECTRAL MINIMAL PARTITIONS OF METRIC GRAPHS 27 [BNH17] V. Bonnaillie-No¨el and B. Helffer,
Nodal and spectral minimal partitions – The state of the art in 2016 ,in A. Henrot (ed.),
Shape optimization and spectral theory , De Gruyter Open, Warsaw-Berlin, 2017.[CTV05] M. Conti, S. Terracini, and G. Verzini,
On a class of optimal partition problems related to the Fuˇc´ıkspectrum and to the monotonicity formulae , Calc. Var. (2005), 45–72.[CW05] S. Currie and B. A. Watson, Dirichlet-Neumann bracketing for boundary-value problems on graphs ,Elec. J. Diff. Eq. (2005), No. 93, 1–11.[Dev89] R. Devaney,
An Introduction to Chaotic Dynamical Systems , 2nd edition, Addison–Wesley, RedwoodCity, 1989.[Fri05] L. Friedlander,
Extremal properties of eigenvalues for a metric graph , Ann. Inst. Fourier (2005),199–212.[KKLM20] J. B. Kennedy, P. Kurasov, C. L´ena, and D. Mugnolo, A theory of spectral partitions of metric graphs ,preprint (2020), arXiv:2005.01126.[KKMM16] J. B. Kennedy, P. Kurasov, G. Malenov´a and D. Mugnolo,
On the spectral gap of a quantum graph ,Ann. Henri Poincar´e (2016), 2439–2473.[Kur08] P. Kurasov, Schr¨odinger operators on graphs and geometry I: Essentially bounded potentials , J. Funct.Anal. (2008), 934–953.[KS18] P. Kurasov and A. Serio,
On the sharpness of spectral estimates for graph Laplacians , Rep. Math. Phys. (2018), 63–80.[Mug19] D. Mugnolo, What is actually a metric graph? , preprint (2019), arXiv:1912.07549.[Nic87] S. Nicaise,
Spectre des r´eseaux topologiques finis , Bull. Sci. Math., II. S´er. (1987), 401–413.[OS19] A. Odˇzak and L. ˇS´ceta,
On the Weyl law for quantum graphs , Bull. Malays. Math. Sci. Soc. (2019),119–131. Matthias Hofmann, Grupo de F´ısica Matem´atica, Faculdade de Ciˆencias, Universidade de Lisboa,Campo Grande, Edif´ıcio C6, P-1749-016 Lisboa, Portugal
E-mail address : [email protected] James B. Kennedy, Grupo de F´ısica Matem´atica and
Departamento de Matem´atica, Faculdadede Ciˆencias, Universidade de Lisboa, Campo Grande, Edif´ıcio C6, P-1749-016 Lisboa, Portugal
E-mail address : [email protected] Delio Mugnolo, Lehrgebiet Analysis, Fakult¨at Mathematik und Informatik, FernUniversit¨atin Hagen, D-58084 Hagen, Germany
E-mail address : [email protected] Marvin Pl¨umer, Lehrgebiet Analysis, Fakult¨at Mathematik und Informatik, FernUniversit¨atin Hagen, D-58084 Hagen, Germany
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