CONVERSE SPECTRAL PROBLEMS FOR NODAL DOMAINS 1 ...
MOSCOW MATHEMATICAL JOURNAL Volume 7, Number 1, January–March 2007, Pages 67–84
CONVERSE SPECTRAL PROBLEMS FOR NODAL DOMAINS B. HELFFER AND T. HOFFMANN-OSTENHOF
Abstract. We consider two-dimensional Schr¨ odinger operators in bounded domains. Abstractions of nodal sets are introduced and spectral conditions for them ensuring that they are actually zero sets of eigenfunctions are given. This is illustrated by an application to optimal partitions. 2000 Math. Subj. Class. 35B05. Key words and phrases. Schr¨ odinger operator, Nodal domain, Spectral theory.
1. Introduction Consider a Schr¨ odinger operator H = −∆ + V
(1.1)
2
on a bounded domain Ω ⊂ R with Dirichlet boundary condition. We assume that ∂Ω has finitely many piecewise smooth components and satisfies an interior and an exterior cone condition. Furthermore we assume that V ∈ C ∞ (Ω) is real valued. The operator H is then selfadjoint if viewed as the Friedrichs extension of the quadratic form associated to H with form domain H01 (Ω). We denote H, by H(Ω). We know that H(Ω) has compact resolvent. So the spectrum of H(Ω), σ(H(Ω)) can be described by an increasing sequence of discrete eigenvalues λ 1 < λ2 6 λ 3 6 · · · 6 λ n 6 · · · tending to +∞, such that the associated eigenfunctions uk can be chosen to form an orthonormal basis in L2 (Ω). We can assume that the eigenfunctions uk are real valued and by elliptic regularity [9] we have uk ∈ C ∞ (Ω) ∩ C00 (Ω).
(1.2)
It is well known that u1 can be chosen to be strictly positive and that the other eigenfunctions uk , k > 1, must have nonempty zero sets. We define the zeroset Received April 19, 2006. The two authors are supported by the European Research Network ‘Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems’ with contract number HPRNCT-2002-00277, and the ESF Scientific Programme in Spectral Theory and Partial Differential Equations (SPECT). c
2007 Independent University of Moscow
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N (u) of a function u by N (u) = {x ∈ Ω : u(x) = 0}.
(1.3)
The nodal domains of an eigenfunction u, which are by definition the connected components of Ω\N (u), will be denoted by Dj , j = 1, . . . , µ(u), where µ(u) denotes the number of nodal domains of u. For any open subset D ⊂ Ω we denote by H(D) the operator −∆ + V with form domain Q(H) = H01 (D). In [2] we considered together with A. Ancona the following situation. Supposing the Ωℓ are pairwise disjoint open subset of Ω, we derived inequalities relating the sum of the spectral counting functions of the H(Ωℓ ) with the counting function of H(Ω). Here the counting function attached to λ and H(Ωℓ ) is the number of eigenvalues of H(Ωℓ ) that are smaller or equal to a given λ. Also converse results were obtained. Namely for the case of equality it turned out that these Ωℓ already must be nodal domains or union of nodal domains. The problem we address here is related in spirit. Let D be a partition of Ω, that is a family of µ connected open subsets of Ω such that ! µ [ Di ∩ Dj = ∅ for i 6= j, and Int Di = Ω. (1.4) 1
If the µ domains happen to be the nodal domains of an eigenfunction u of H(Ω) such that H(Ω)u = λu, then λ is in the spectrum of any operator HI where I is a subset of {1, 2, . . . , µ} and !! [ HI = H Int Di . (1.5) i∈I
We are interested in the possibility of a converse statement. If we look first at the one dimensional case when Ω is an interval in R, we easily see that, if we can find a λ ∈ R such that λ is a groundstate energy for each H(Di ), then this λ should be an eigenvalue of H(Ω) and we can find a corresponding eigenfunction u such that the intervals Di are the nodal domains of u. We have just indeed to multiply each eigenfunction ui of H(Di ) by a constant ci in order to glue them together for getting an eigenfunction u. In order to go further, we first observe that this is no longer true in the case of a circle S 1 . Some compatibility condition should occur and it is rather easy to find examples for which one cannot glue together the ui ’s. As we shall see later, this phenomenon could also appear in higher dimension when Ω is not simply connected. The second observation is that this gluing procedure does not work anymore in higher dimension. In order to explain the problem, let us first give a definition. We say that two open sets Di , Dj of D are neighbors, or neighbor each other, or Di ∼ Dj if Dij := Int(Di ∪ Dj ) is connected. (1.6) Now if for two neighbors Di and Dj , λ is the groundstate energy of both H(Di ) and H(Dj ), there is no way in general to construct uij in the domain of H(Dij ) such that uij = ci ui in Di and uij = cj uj in Dj . We would indeed need at ∂Di ∩ ∂Dj the normal derivatives of ui and uj to be proportional.
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So it is natural for the analysis of the converse problem to assume in higher dimension the existence of uij for all the pairs of neighbouring domains and to try then to glue those uij so that an eigenfunction of the whole problem is obtained. We are consequently led to the following definition: Definition 1.1. We say that the partition D = {D1 , . . . , Dµ } satisfies the pair compatibility condition, for short (PCC), if, for some λ ∈ R, and for any pair (i, j) such that Di ∼ Dj , there is an eigenfunction uij 6≡ 0 of H(Dij ) such that H(Dij )uij = λuij with N (uij ) = ∂Di ∩ ∂Dj .
(1.7)
We can associate (see Figure 1 for an example) to such a partition D a graph G or G(D) by placing in each Di , i = 1, . . . , µ, a vertex vi and by associating edges eij to the vi , vj such that the corresponding Di , Dj satisfy Di ∼ Dj . We say that D is admissible if the associated graph G(D) is bipartite. Bipartite graphs are just graphs whose vertices can be colored by two colors so that vertices which are joined by an edge have different colors. This is a well known notion in graph theory, see e. g. Diestel [8]. We shall see in the next section that the nodal set N (u) of an eigenfunction u of H(Ω) has, under the condition that ∂Ω is C ∞ , the following abstract nodal set property (or shortly (ANSP)) which we now define. e (Ω) Definition 1.2 (abstract nodal set property). A closed set N in Ω belongs to N if N meets the following requirements:
(i) N is the union of smooth arcs connecting points in ∂Ω and smoothly imbedded circles in Ω. (ii) There are finitely many distinct xi ∈ Ω∩N and associated positive integers νi (νi > 2) such that, in a sufficiently small neighborhood of each of the xi , N is the union of ν(xi ) C ∞ curves (non self-crossing) two by two crossing transversally at xi (with positive angle) and such that in the complement of these points in Ω, N is locally diffeomorphic to a regular curve. (iii) ∂Ω ∩ N consists of a (possibly empty) finite set of points zj , such that, at each zj , ρj (ρj > 1) nodal lines hit the boundary. Moreover, for each zj ∈ ∂Ω, assuming that we have rotated and translated Ω such that zj = {0}, that ∂Ω is at the origin tangent to the x1 -axis and that Ω lies locally above the x1 -axis, then N is near zj the union of ρj distinct C ∞ half-curves which hit the origin with strictly positive angles.
By smooth we mean as usual that each arc, respectively circle, is a component of the zeroset of a C ∞ function which has at the zero nonvanishing gradient. The points xi will be called “critical” points of the (abstract) nodal set. Let us also observe that this definition implies that the family of abstract nodal domains which are by definition the components of Ω\ N is an admissible partition. Conversely, we can associate to a partition D the closed set [ N (D) = ∂Di \ ∂Ω, i
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5 1
2
10
3
4
6 9
7
8
1 7
2
5
3 4 6
8
10 9
Figure 1. Partition and associated graph and will always assume that the partition is nice in the sense that [ Di . Ω \ N (D) =
(1.8)
i
Our main theorem is the following
Theorem 1.3. Suppose that Ω is simply connected with smooth boundary and that, e (Ω) and λ ∈ R, the associated family D = {D1 , . . . , Dµ } satisfies for some N ∈ N (PCC). Then there is an eigenfunction of H(Ω) with corresponding eigenvalue λ such that N (D) = N (u). Remarks 1.4. (i) The regularity assumptions can probably be relaxed, but we do not strive for generality here. See [2] for the type of conditions which could be given.
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(ii) If Ω is not simply connected then our result does not hold in general as will be explained through examples in Section 6. There we also give additional conditions on the admissible partition D such that Theorem 1.3 still holds. (iii) There are simple cases in higher dimensions for which Theorem 1.3 easily can be shown to hold (for example in some simple tree situations). This is a natural question to ask if, beyond the interest to have a theoretical criterion with minimal compatibility condition, one can check more concretely the assumptions of the main theorem. The recent contributions of Conti–Terracini– Verzini [4], [5], [6] give an interesting application1. The pair compatibility condition will appear naturally when, given an integer k and an open set Ω, one considers partitions of Ω by k open sets Di . One is then interested in the properties of optimal partitions, i. e. partitions by k open sets Di , for which maxi (λ(Di )) is minimal. A natural question is to determine when an optimal partition corresponds to a nodal pattern. As an application of our main theorem, we obtain: Corollary 1.5. Let Ω be simply connected, k ∈ N (k > 2) and let D min = (Di )i=1,...,k be a optimal admissible nice (in the sense of (1.8)) partition, such e (Ω). Then there is an eigenfunction u of H(Ω) associated with that N (D min ) ∈ N λ = max(λ(Di )), i
such that D
min
is the family of the k nodal domains of u.
Proof. Let us apply Theorem 1.3. We take as λ = maxi (λ(Di )). The first point is that all the λ(Di ) should be equal. If not, one could by deformation of the Di in a neighborhood of regular points of their boundary find a e which would decrease maxi (λ(Di )). new partition D, The second point is to observe that considering two neighbors Di and Dj , then λ should be the second eigenvalue of H(Dij ). If it was not the case for some pair (i, j), the two nodal domains of the second eigenfunction of H(Dij ) will give two new open sets Di′ and Dj′ with λ(Di′ ) = λ(Dj′ ), in contradiction with the assumption of optimality and the first point of the proof. Hence the pair compatibility condition is satisfied. Remarks 1.6. (i) The case where k = 2 corresponds to a rather well known characterization of the second eigenvalue of H(Ω). The admissibility condition is of course automatically satisfied in this case. (ii) In their sequence of papers [4], [5], [6], Conti–Terracini–Verzini, partly motivated by questions in biomathematics, have shown that there exists at least one optimal partition with these properties. P These authors consider first the minimizations of expressions of the type i (λ(Di ))p , for p ∈ (1, +∞) and the case considered here appears as a limiting (in some sense more difficult) case. (iii) In Corollary 1.5, the assumptions of regularity (and may be also the assumption that Ω is simply connected) can hopefully be removed (cf [11]), but the assumption that the partition is admissible is essential. It is easy to find examples 1We thank M. van den Berg for mentioning to us one of these articles and S. Terracini for useful discussions about these questions of optimal partitions.
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(take the disk and partitions by three open sets) where the optimal partitions do not correspond to a family of nodal domains! Organization of the paper. In Section 2 we collect some well known facts about zero sets and nodal domains. In Section 3, we analyze the properties of partitions of Ω in connection with graph theory. Section 4 is devoted to the proof of the main theorem. In Sections 5, we illustrate the theorem by discussing examples. Section 6 is devoted to a general criterion for non simply connected domains. We then analyze in Section 7 the optimality of these sufficient conditions by considering families of examples for which (PCC) does not imply a general compatibility condition. Concluding remarks are given in Section 8. 2. Regularity of Eigenfunctions and Abstract Nodal Set Property We investigate the properties of nodal domains and nodal sets. This will lead us to propose and justify the corresponding abstractions. In particular we will show that for a smooth Ω the zerosets of the eigenfunctions satisfy the abstract nodal set property introduced in Definition 1.2. First we recall some basic regularity results (cf. [9]). Proposition 2.1. Every eigenfunction u of H(Ω) belongs to C ∞ (Ω) ∩ C00 (Ω). Furthermore, for any eigenfunction u, any nodal domain ∂D is piecewise smooth and satisfies an interior cone condition. If in addition the boundary is C ∞ then u ∈ C ∞ (Ω). e (Ω). The next property justifies the introduction of N
Proposition 2.2. Suppose ∂Ω ∈ C ∞ and u is an eigenfunction of H(Ω). Then e (Ω). N (u) belongs to N Moreover the nodal lines can only cross at interior points with equal angles, and at boundary points crossing nodal half-lines determine also together with the boundary equal angles. The proof follows rather directly from the local behaviour of eigenfunctions near their zeros. Lemma 2.3. Let u in H01 (Ω∩B(x0 , ρ0 )) with Ω∩B(x0 , ρ0 ) 6= ∅ where B(x0 , ρ0 ) = {x ∈ R2 : |x − x0 | < ρ0 } and x0 ∈ Ω. Let us assume that: (−∆ + V − λ)u = 0
in Ω ∩ B(x0 , ρ0 ).
(a) Suppose x0 ∈ N (u) ∩ Ω then there is ν such that, u(x) = Pν (x − x0 ) + Pν+1 (x − x0 ) + O(|x − x0 |ν+2 ),
(2.1)
in a sufficiently small neighborhood of x0 . Thereby Pν 6≡ 0, defined by Pν (x) = r ν (a cos νω + b sin νω),
(2.2)
is a harmonic homogeneous polynomial of degree ν. For simplicity we have written this in polar coordinates r, ω. Note that the zeroset of any Pm consists of m straight lines which intersect with equal angles.
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(b) If z0 ∈ N (u)∩∂Ω then there exist ρ > 1 and r0 > 0 such that, in Ω∩B(z0 , r0 ), u(x) = Pρ+1 (x − z0 ) + O(|x − z0 |ρ+2 ).
(2.3)
Thereby Pρ+1 6≡ 0 in (2.3) is defined as in (2.2) and has the property that the line tangent to ∂Ω at z0 is in the zeroset of Pρ+1 (x − z0 ). About the proof of Lemma 2.3. This lemma is well known among specialists (see for instance [3] or [12]). A detailed proof of the boundary case is for example given in [10] using the reflection argument along the lines of [13]. Remarks 2.4. (i) Note that for each component of ∂Ω the number of nodal lines hitting this component has to be even. This is implied by the property that the graph associated to N is bipartite. Indeed, otherwise the associated graph would contain an “odd circle” (that is a circle with an odd number of vertices) and this would make it impossible to color the associated graph with two colors. (ii) It is useful to have also a description of the zeros of eigenfunctions in the case of domains with piecewise C ∞ boundaries and to describe the local structure of the zeros near the corners. This is discussed for example in [7]. The main difficulty for having a local structure lemma at the corner is to show that the solution cannot be flat at the corner, that is cannot decay faster than polynomially. This can be proved2 (see [1] or [11]) by the use of a conformal transformation. 3. Graphs and Circulation Along Paths 3.1. Preliminaries. We start with a partition D = {D1 , . . . , Dµ } so that [ e (Ω). N= ∂Di \ ∂Ω ∈ N
We assume that for some given λ, D satisfies (PCC). We introduce various normalizations. For i = 1, . . . , µ, let ui be the positive normalized groundstate of H(Di ). Similarly, for any oriented pair (i, j) such that Di and Dj are neighbors, let uij be the eigenfunction of H(Dij ) having ∂Di ∩ ∂Dj as nodal set, hence Di and Dj as nodal domains. We normalize uij and impose that uij is strictly positive in Di . Then uij is uniquely determined and negative in Dj . Note that with this choice uij = −uji , (3.1) and that we can write uij = dij (ui − γij uj ),
(3.2)
with γij and dij strictly positive. We can then write γij = exp cij , and we call cij the circulation from Di to Dj . Having in mind (3.1), we get the relations : dij = dji γji ,
dij γij = dji .
2We thank S. Terracini for explaining to us the proof.
(3.3)
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In particular we get the important relation: γij γji = 1,
(3.4)
which in the circulation terminology becomes cij + cji = 0.
(3.5)
3.2. Good paths. We now consider continuous paths in Ω wandering between the nodal domains. More precisely the following notion of good path is useful: Definition 3.1. We say that the path [0, 1] ∋ t 7→ β(t) ∈ Ω is good (with respect to D) if (i) β(0), β(1) ∈ Ω \ N , β(t) ∩ ∂Ω = ∅ for t ∈ [0, 1]. (ii) β(t) ∩ N (ν) = ∅ for ν > 2 where N (ν) = {x ∈ N ∩ Ω : ν nodal lines locally cross in x}. (iii) If for some t0 ∈ (0, 1), β(t0 ) ∈ ∂Di ∩ ∂Dj , then there is an ǫ > 0 such that, for t ∈ (t0 − ǫ, t0 ), β(t) ∈ Di (or Dj ) and for t0 ∈ (t0 , t0 + ǫ), β(t) ∈ Dj (or Di ). To any good path, we can associate a finite sequence i0 , i1 , . . . , ik of indices expressing the restriction of the path to the graph G. We call this restriction the associated G-path and denote it by βG . This simply means that the path starts from β(0) ∈ Di0 , then leaves Di0 for entering in Di1 and a new index is added at each crossing of a boundary. The length of the path is then exactly the number of crossings of the path. As usual, we say that the path is closed if β(0) = β(1). Note that if β(0) and β(1) belongs to the same Di then we can always close the path (using the property that Di is arcwise connected), keeping the corresponding graph fixed, which is in any case a circle. 3.3. Circulation along βG . We can associate to each good path β(t) two numbers γβ =
k−1 Y
γiℓ ,iℓ+1 ,
k−1 X
ciℓ ,iℓ+1 ,
ℓ=0
and Cβ =
ℓ=0
the second one being called the circulation along β. Of course, we have γ β = exp Cβ . Note that Cβ depends only on the G-path βG , so Cβ will also be called the circulation along βG . When we deform these paths by homotopy, it is clear that as long as the path keeps the property of being good the circulation is constant. But one of our goals will be to follow this circulation when changing in the homotopy the corresponding G-path (that is the path in the homotopy does not remain a good path).
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4. Proof of the Main Theorem 4.1. General compatibility condition. The general condition for constructing an eigenfunction of H(Ω) by gluing the uij of H(Dij ) is quite reminiscent of the problems occuring in Cech Cohomology3 but we have finally preferred to remain selfcontained, instead of sending the reader to a too sophisticated book in cohomology. The following criterion4 is quite natural: Proposition 4.1. If we have an admissible partition D in Ω generated by some N e (Ω) such that (PCC) holds for some λ ∈ R, then an eigenfunction of H(Ω) in N associated with λ can be constructed if and only if, for any closed path on the graph of length k > 2, the condition (GCC): k−1 Y
γiℓ ,iℓ+1 = 1,
(4.1)
ℓ=0
is satisfied.
In other words, the circulation along any closed path on the graph must be 0. Remarks 4.2. (i) At this stage, it is not necessary to assume that Ω is simply connected. (ii) Note that because we are in a bipartite graph, k has to be even in (4.1). Proof of Proposition 4.1. One starts from one domain Di0 and from its groundstate ui0 . Then the extension of ui0 to all neighboring domains is obtained by using assumption (PCC). One can then propagate the extension to the next neighbors till Ω is covered. (GCC) just permits a construction which is independent of the path used for the extension. This gives a global construction of an element u in 2 H01 (Ω) belonging to Hloc (Ω \ N c ) where N c is the set of critical points of N lying in Ω, and satisfying in the distribution sense (−∆ + V − λ)u = 0 in Ω \ N c . But (−∆ + V − λ)u belongs to H−1 (Ω) so it remains to show that a distribution in H−1 with support in N c is 0. But N c is a finite set and it is enough to recall the standard result in distribution theory: Lemma 4.3. If ω is an open set in R2 and x0 ∈ ω, there are no distribution in −1 Hloc (ω) with support in {x0 }. This achieves the proof of the proposition.
3Note however that we are not working with coverings but with partitions and that the γ ij are only defined for pairs of neighbors in the specific sense given in the introduction. So the γij ’s define a cochain on Ω′ corresponding to Ω \ ∪i xi , where the xi ’s denote the singular points of the abstract nodal set. 4 In the algebraic topology language, the criterion says essentially that, when our cochain is a cocycle, it is a coboundary.
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So the proof of our main theorem consists in showing that Condition (4.1) is always satisfied when Ω is simply connected. In particular, it is not too difficult to see that the condition is automatically satisfied in the case when the graph associated to the partition is a tree. 4.2. Further reduction. We now explain how to reduce the computation of the circulation along a good closed path to the case where a given good path encloses only one or no critical points. The proof is by induction. Suppose that a closed good path γ parametrized by t ∈ [0, 1] encloses exactly k critical points (with k > 2). The claim is that we can find two points (inside nodal domains) on this closed path corresponding to times t0 and t1 , and construct a continuous curve ℓ01 going from γ(t0 ) to γ(t1 ) avoiding the critical set such that the path γ1 defined by γ1 = γ on [0, t0 ], by ℓ01 on [t0 , t1 ] (after reparametrization) and by γ1 = γ on [t1 , 1] is a good path containing only in its interior one critical point. If now ℓ10 denotes the opposite path to ℓ01 , we can consider the closed good path γ2 such that γ2 = ℓ10 on [0, t0 ], γ2 = γ on [t0 , t1 ]. It is clear that γ2 encloses in its interior (k − 1) critical points and that the circulation along γ is the sum of the circulation along γ1 and of the circulation along γ2 . If (k − 1) > 2, we can iterate the procedure till each path encloses at most one critical point. So the general proof is reduced to the analysis of the condition (4.1) in the case when a good closed path either encloses no critical point or one critical point. This will be the object of the two next subsections. We will actually show that in these two situations the circulation along the path is zero, when the path is homotopic in Ω to a point, which is automatically the case if Ω is simply connected. 4.3. Proof when a good closed path does not enclose any critical point and is homotopic to a point. In this case we can find a homotopy γ(s, t) such that γ(0, t) is the initial path, γ(1, t) is a single point living in some Di . Of course the graph trace of the path, i. e. βG , is changing with this homotopy. But since each nodal domain has only finitely many critical points in its boundary a continuity argument shows that, by modifying the homotopy, we can get one for which there are only finitely many sk for which the paths are no more good paths. Moreover one can pick this homotopy in such a way that, at these sk , the paths γ(sk , ·) have still the property that they are good except at one point tk . So the transition near the point (denoted by S0 in the figure) γ(sk , tk ) is the following (or the converse). There exists some pair of neighboring Di and Dj such that γ(sk , tk ) ∈ ∂Di ∩ ∂Dj . For s < sk , with (s, t) near (sk , tk ), the path γ(s, t) (see Figure 2) is contained in Diℓ , with iℓ = i. For s = sk and t near tk , γ(sk , t) belongs to Diℓ except at γ(sk , tk ). For s > sk with s near sk , the path enters the neighbor Dj = Dinew , before returning to Diℓ and entering Diℓ+1 . In particular we can pick ℓ+1 the homotopy always so that the path avoids any critical point. So the initial corresponding G-path i1 , i2 , i3 , . . . , iℓ−1 , iℓ , iℓ+1 , . . . , ik becomes i1 , i2 , i3 , . . . , iℓ−1 , iℓ , inew ℓ+1 , iℓ , iℓ+1 , . . . , ik . The fact that the circulation is conserved in this transformation is an immediate consequence of (3.4).
CONVERSE SPECTRAL PROBLEMS FOR NODAL DOMAINS
D il+1 Di l
D il+1
D il+1 S0
77
D
Dinew l+1
new il+1
S0
Di l
Di l
S0
Dinew l+1
Figure 2. Deformation argument in three pictures: before, at the touching and after
For the converse transition, we just replace a sequence iℓ , inew ℓ+1 , iℓ , iℓ+1 by iℓ . After finitely many operations of this type, we will obtain a path reduced to a single point whose G-path is also a point. Actually the main point here is that the associated closed G-path is a path on a tree! 4.4. Proof when a good closed path encloses a unique critical point and is homotopic to a point. Let us now consider the case of a closed path encloses a unique critical point. We can reduce the computation to the case when this closed good path is a small circle turning once and positively around this point x0 . There exists ν > 2 so that the ν nodal arcs σ0 , σ1 , . . . , σν−1 pass locally through x0 . This means that there is an ǫ0 > 0 such that for 0 < ǫ < ǫ0 N ∩ B(x0 , ǫ) =
ν−1 [
σj ∩ B(x0 , ǫ)
j=0
where B(x0 , ǫ) = {x ∈ R2 : |x − x0 | < ǫ}. We shall also use the (2ν) half arcs σℓ+ + (such that σℓ = σℓ+ ∪ σℓ+ν ). Without loss we might assume that x0 = {0} and that σ0 is tangent to the x1 -axis at x0 . The arc σ0 splits B(x0 , ǫ) in two parts and we denote by B + (x0 , ǫ) the upper part which lies “above” σ0 . σ1
σ2
σ1
+
+ σ2
S3 σ3+
+
B(x0,ε)
S2
x0
S6
S4 S5 σ4+
σ0 + + σ0 = σ6
S1
σ5+
Figure 3. Picture in the case when ν = 3
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Assume now (PCC) for D. The abstract nodal set (intersected with B(x0 , ǫ)) defines 2ν curved sectors Sℓ (also intersected with B(x0 , ǫ))5 (ℓ = 1, . . . , 2ν) de+ limited6 by σℓ−1 and σℓ+ , each one belonging to some Diℓ . Take the first domain S1 starting from σ0+ . Starting from ui1 , the pair compatibility condition can be used iteratively to extend the restriction of ui1 to S1 as a local solution v1 ∈ C ∞ (B + (x0 , ǫ)) such that (−∆ + V )v1 = λv1 in B + (x0 , ǫ), v1 = 0 for N ∩ B + (x0 , ǫ) and v1 = ui1 on S1 . We can now apply Lemma 2.3 (b) to v1 in B + so that7 v1 = c1 r ν sin νω + O(r ν+1 ) for some constant c1 6= 0. In particular this means that though we have not assumed that the nodal lines cross at the point x0 under equal angles this is enforced by Lemma 2.3. The second point is that by restriction to S1 , we get ui1 = c1 r ν sin νω + O(r ν+1 ) in S1 . and a similar expansion is true for ui1 i2 : ui1 i2 = c12 r ν sin νω + O(r ν+1 )
in Int(S 1 ∪ S 2 ).
Of course we can do the same thing starting from any Sℓ . So, for each ℓ, we have shown the existence of cℓ > 0 such that uiℓ has the asymptotics uiℓ = −cℓ r ν sin (ν(ω − ωℓ,ℓ+1 )) + O(r ν+1 ),
(4.2)
in Sℓ , where ωℓ,ℓ+1 is the argument of the tangent to ∂Sℓ ∩ ∂Sℓ+1 at 0 and a similar expansion holds for uiℓ iℓ+1 . But reusing (PCC) (through (3.2)) gives that cℓ+1 . (4.3) γiℓ ,iℓ+1 = cℓ Coming back to the definition of the circulation and using (4.3) we get the vanishing of the circulation along the good path enclosing the critical point. This completes the consideration of this case and finishes also the proof of the main theorem. 5. Examples We apply the general constructions above for the analysis of specific examples. 5.1. Three examples whose corresponding graph is a tree A simple partition. The left subfigure in Figure 4 presents a partition by five domains, whose corresponding graph is a tree. Moreover, there are no critical points inside the domain. 5We omit from now on recalling the fact that we are always in a small ball around x . But 0
the whole proof is local. See Figure 3 for the picture in the case ν = 3. 6By convention, σ + is σ + . 2ν 0 7Here there are two possibilities in the choice of polar coordinates. We can either flatten the boundary of B + and take polar coordinates after flattening or keep the initial ones. In any case, the two choices lead to the same main term.
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The eight. Take the case of the eight as in the central subfigure of Figure 4 with (0, 0) as critical point. Let D2 and D3 the two interior “abstract” nodal domains and D1 the “exterior” nodal domain meeting ∂Ω. Then D1 ∼ D2 and D1 ∼ D3 . Suppose that, for some λ, we have (PCC). We apply Lemma 2.3 (case (b)) locally r0 ,± r,± to u12 in D12 , (r0 > 0 small enough), where D12 = Int(D2r ∪ D1r,± ), D1r,± being the two components (for r small enough) of D1 ∩ B(0, r). This shows that the groundstate u1 (which is up to a multiplicative constant the restriction to D1 of u12 ) satisfies u1 = c1 r 2 sin(2(ω − ω± )) + O(r 3 ), near x0 = 0, where the constant c1 is the same for the two “opposite” sectors describing D1 near x0 . But the associate graph is a tree. It is trivial in this case that the trace of a good closed path on the graph has always zero circulation. We do not need to use the information given by the local analysis around the critical point. The only additional information given by this analysis is that u1 has the same asymptotics near 0 in the two opposite sectors. The clover leaf intersection. This example (right subfigure in Figure 4) does not lead to any difficulty. The graph is a tree. On can directly extend from D1 toward respectively D2 , D3 and D4 .
1 4
2
3
5
1
1 2
3
3 4
2
1 5
4
1
1
3 2
2
3
2
3
4
Figure 4. Three examples with associated graph below: simple, eight and clover
5.2. Examples with circles The cross. When Ω is the disk B(0, 1), the cross (say {x1 = 0} ∪ {x2 = 0} determines four nodal domains Dj (j = 1, 2, 3, 4) (see the left subfigure in Figure 5 and its corresponding graph below) so we have a “circle” (1, 2, 3, 4). The corresponding graph can be represented by a square. Here we cannot avoid the local analysis around the center.
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1 1
4
4 3
2
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1
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1
4
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3
Figure 5. Two other examples in simply connected domains with square graph Two intersecting circles. Although the corresponding graph is the same, the nodal structure is different. There are two critical points. There are three typical paths enclosing critical points. One is living in D1 and turning positively and once around D2 ∪D3 ∪D4 . By definition, the circulation along this path is trivial. The second one is a small positively oriented circle around the left critical point. Its corresponding trace in the graph is the sequence (1, 2, 3, 4). Here we need to perform the local analysis. The third one is a small positively oriented circle around the right critical point. Its corresponding trace in the graph is the sequence (4, 3, 2, 1). One can perform the local analysis but also observe that the circulation along this path is just the opposite of the previous one. 6. Sufficient Conditions in the Non Simply Connected Case In the non simply connected case, what remains from the previous proof can be formulated as follows. Proposition 6.1. In each homotopy class of Ω, all the good paths have the same circulation. In particular we can speak of a circulation attached to a homotopy class. Remark 6.2. As a consequence of Proposition 6.1, for a given partition satisfying (ANSP) and (PCC), the proof that (GCC) holds is reduced to the proof that in each homotopy class there is a representative with circulation 0. Of course we recover in the simply connected case Theorem 1.3. Remark 6.3. It is actually enough (using the properties of the fundamental group of Ω) to verify (GCC) for a set of generators of this group.
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In this spirit the case of one hole can be treated in greater detail. The homotopy group is generated by the (class of) simple path (s) turning once and anticlockwise around the hole. This leads to the following sufficient condition: Proposition 6.4. We assume that Ω has just one hole and that, for some partition D and λ, (PCC) is satisfied. If there is a good path of index 1 around the hole, which intersects the abstract nodal set at at most two points, then the conclusions of Theorem 1.3 hold. The proof is immediate using Proposition 6.1, Remark 6.2 and the fact that the circulation along a good path with no crossing is 0 by definition and is also 0 in the case of two crossings by (PCC). Typically the assumptions are satisfied when the hole or the exterior boundary are hit by no or two nodal lines. 7. On the Optimality in the Non Simply Connected Case Let us consider the case with one hole and let λ be an eigenvalue of multiplicity 1 such that the corresponding eigenfunction u has at least four nodal domains. We would like to present a family of examples for which one can then construct a new potential so that the main theorem does not hold any more in spite of (PCC). Assumption 7.1. We assume that there exists one nodal domain D such that Ω \ D becomes simply connected and such that the boundary of D with each of its neighbors is connected. We also assume that ∂D ∩ ∂Ω meets the regular parts of the exterior boundary and of the interior boundary. Then the claim is Proposition 7.2. Under the previous assumptions, we can find a new potential Vǫ such that λ satisfies the pair compatibility condition corresponding to N (u) and such that λ is not an eigenvalue of −∆ + Vǫ with an eigenfunction having the same nodal domain as u. The proof is inspired by the analysis of the case of the circle presented in the introduction. By assumption ∂D contains two distinct non crossing continuous curves L± joining the two boundaries. We proceed by constructing a C ∞ function b in D, such that the support of ∇b does not meet L± , b = 0 near L+ , and b = b− near L− , where b− ∈ R \ {0}. Moreover, we can require that ∇b · n = 0 on ∂Ω ∩ ∂D,
(7.1)
where n is the outward normal to ∂Ω. We note that b can be extended by 0 outside D to Ω \ L− but not to Ω. On the contrary, ∇b and ∆b can be extended to the whole Ω! We now introduce uǫ = (1 + ǫb)u. (7.2)
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L-
b=bb=0 L+ supp ∇b
Figure 6. Construction of b in the simplest case We note that uǫ is well defined in Ω \ L− and can also admit an extension, when crossing L− , by (1 + ǫb− )u to the neighbors of D touching D in L− . Moreover, if ǫ 6= 0, it cannot be extended to a C ∞ function in Ω. We now claim that, for small enough ǫ 6= 0, uǫ is an eigenfunction of −∆ + Vǫ in Ω \ L− , with eigenvalue λ and with ∆b ∇b · ∇u Vǫ := V + ǫ +2 . (7.3) 1 + ǫb uǫ We observe, using the property (7.1) and that ∇u is not vanishing on the boundary on the support of ∇b, by Hopf’s boundary point Lemma (see [9]), that Vǫ admits a C ∞ extension to Ω. Now (PCC) is satisfied for λ, −∆ + Vǫ in Ω and the family associated to N (u). If λ was an eigenvalue of the Dirichlet realization of −∆+Vǫ in Ω with a corresponding eigenfunction vǫ with nodal set N (u), then comparing vǫ and uǫ in D, we would get vǫ = cǫ uǫ . Remark 7.3. Note that we do not know if λ is an eigenvalue of the Dirichlet realization of −∆ + Vǫ in Ω. But vǫ is C ∞ in Ω and uǫ has a discontinuity! Hence a contradiction. Remarks 7.4. (i) More general families of examples are treated in [10]. (ii) One could ask naturally if replacing (PCC) by the triple compatibility condition (TCC) will lead to other results. Using the same ideas as above, it is easy to construct examples for which (TCC) does not imply (GCC). 8. Final Remarks In this paper we have analyzed some of the properties (local, global, spectral) satisfied by a family of sets formed by nodal domains of an eigenfunction.
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We have then proposed a sufficient natural pair compatibility condition permitting to glue together eigenfunctions attached to each pair of neighboring domains. We have shown its sufficiency in the case when Ω is a simply connected open set in R2 and described how one can extend the analysis in the non simply connected situation. The analysis of a family of examples shows that the sufficient conditions we have proposed are in some sense not far from optimal. Except trivial cases, where no circle in the corresponding graph can occur, the analysis of the same question in dimension > 2 is completely open. A precise description of the structure of the nodal set of the eigenfunction near its critical points is indeed missing. Let us finally mention that the same problem can be considered for a Schr¨ odinger operator on a surface either with or without boundary. Then the genus has to e (Ω) play a role. While for the flat case and actually also for the sphere N ∈ N automatically guarantees that the nodal domains created by Ω \ N lead to an admissible family D this has to be required for the case of surfaces in general. Take for instance the torus: then an N which is just a simple closed loop which is not zero-homotopic creates just one nodal domain, hence not an admissible D. Acknowledgements. We thank S. Alinhac, M. Dauge, S. Fournais, H. Koch, L. Robbiano and T. Ramond for help or discussions around Strong Uniqueness theorems and various aspects of the theory of boundary value problems in domains with corners. We thank also M. van den Berg and S. Terracini for instructive discussions about the question of optimal partition. References [1] G. Alessandrini, Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains, Comment. Math. Helv. 69 (1994), no. 1, 142–154. MR 1259610 [2] A. Ancona, B. Helffer, and T. Hoffmann-Ostenhof, Nodal domain theorems ` a la Courant, Doc. Math. 9 (2004), 283–299 (electronic). MR 2117417 [3] L. Bers, Local behavior of solutions of general linear elliptic equations, Comm. Pure Appl. Math. 8 (1955), 473–496. MR 0075416 [4] M. Conti, S. Terracini, and G. Verzini, An optimal partition problem related to nonlinear eigenvalues, J. Funct. Anal. 198 (2003), no. 1, 160–196. MR 1962357 [5] M. Conti, S. Terracini, and G. Verzini, On a class of optimal partition problems related to the Fuˇ c´ık spectrum and to the monotonicity formulae, Calc. Var. Partial Differential Equations 22 (2005), no. 1, 45–72. MR 2105968 [6] M. Conti, S. Terracini, and G. Verzini, A variational problem for the spatial segregation of reaction-diffusion systems, Indiana Univ. Math. J. 54 (2005), no. 3, 779–815. MR 2151234 [7] M. Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. MR 961439 [8] R. Diestel, Graph theory, Graduate Texts in Mathematics, vol. 173, Springer-Verlag, New York, 2000. MR 1743598 [9] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190 [10] B. Helffer and M. Hoffmann-Ostenhof, Converse spectral problems for nodal domains, Extended version. Preprint, 2005. [11] B. Helffer, M. Hoffmann-Ostenhof, and S. Terracini, work in progress. [12] T. Hoffmann-Ostenhof, P. W. Michor, and N. Nadirashvili, Bounds on the multiplicity of eigenvalues for fixed membranes, Geom. Funct. Anal. 9 (1999), no. 6, 1169–1188. MR 1736932
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[13] L. Robbiano, Fonction de coˆ ut et contrˆ ole des solutions des ´ equations hyperboliques, Asymptotic Anal. 10 (1995), no. 2, 95–115. MR 1324385 D´ epartement de Math´ ematiques, Bat. 425, Universit´ e Paris-Sud, 91 405 Orsay Cedex, France E-mail address: [email protected] ¨r Theoretische Chemie, Universita ¨t Wien, Wa ¨hringer Strasse 17, A-1090 Institut fu Wien, Austria, and ¨ dinger Institute for Mathematical Physics, BoltzmanInternational Erwin Schro ngasse 9, A-1090 Wien, Austria E-mail address: [email protected]
CONVERSE SPECTRAL PROBLEMS FOR NODAL DOMAINS B. HELFFER AND T. HOFFMANN-OSTENHOF
Abstract. We consider two-dimensional Schr¨ odinger operators in bounded domains. Abstractions of nodal sets are introduced and spectral conditions for them ensuring that they are actually zero sets of eigenfunctions are given. This is illustrated by an application to optimal partitions. 2000 Math. Subj. Class. 35B05. Key words and phrases. Schr¨ odinger operator, Nodal domain, Spectral theory.
1. Introduction Consider a Schr¨ odinger operator H = −∆ + V
(1.1)
2
on a bounded domain Ω ⊂ R with Dirichlet boundary condition. We assume that ∂Ω has finitely many piecewise smooth components and satisfies an interior and an exterior cone condition. Furthermore we assume that V ∈ C ∞ (Ω) is real valued. The operator H is then selfadjoint if viewed as the Friedrichs extension of the quadratic form associated to H with form domain H01 (Ω). We denote H, by H(Ω). We know that H(Ω) has compact resolvent. So the spectrum of H(Ω), σ(H(Ω)) can be described by an increasing sequence of discrete eigenvalues λ 1 < λ2 6 λ 3 6 · · · 6 λ n 6 · · · tending to +∞, such that the associated eigenfunctions uk can be chosen to form an orthonormal basis in L2 (Ω). We can assume that the eigenfunctions uk are real valued and by elliptic regularity [9] we have uk ∈ C ∞ (Ω) ∩ C00 (Ω).
(1.2)
It is well known that u1 can be chosen to be strictly positive and that the other eigenfunctions uk , k > 1, must have nonempty zero sets. We define the zeroset Received April 19, 2006. The two authors are supported by the European Research Network ‘Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems’ with contract number HPRNCT-2002-00277, and the ESF Scientific Programme in Spectral Theory and Partial Differential Equations (SPECT). c
2007 Independent University of Moscow
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N (u) of a function u by N (u) = {x ∈ Ω : u(x) = 0}.
(1.3)
The nodal domains of an eigenfunction u, which are by definition the connected components of Ω\N (u), will be denoted by Dj , j = 1, . . . , µ(u), where µ(u) denotes the number of nodal domains of u. For any open subset D ⊂ Ω we denote by H(D) the operator −∆ + V with form domain Q(H) = H01 (D). In [2] we considered together with A. Ancona the following situation. Supposing the Ωℓ are pairwise disjoint open subset of Ω, we derived inequalities relating the sum of the spectral counting functions of the H(Ωℓ ) with the counting function of H(Ω). Here the counting function attached to λ and H(Ωℓ ) is the number of eigenvalues of H(Ωℓ ) that are smaller or equal to a given λ. Also converse results were obtained. Namely for the case of equality it turned out that these Ωℓ already must be nodal domains or union of nodal domains. The problem we address here is related in spirit. Let D be a partition of Ω, that is a family of µ connected open subsets of Ω such that ! µ [ Di ∩ Dj = ∅ for i 6= j, and Int Di = Ω. (1.4) 1
If the µ domains happen to be the nodal domains of an eigenfunction u of H(Ω) such that H(Ω)u = λu, then λ is in the spectrum of any operator HI where I is a subset of {1, 2, . . . , µ} and !! [ HI = H Int Di . (1.5) i∈I
We are interested in the possibility of a converse statement. If we look first at the one dimensional case when Ω is an interval in R, we easily see that, if we can find a λ ∈ R such that λ is a groundstate energy for each H(Di ), then this λ should be an eigenvalue of H(Ω) and we can find a corresponding eigenfunction u such that the intervals Di are the nodal domains of u. We have just indeed to multiply each eigenfunction ui of H(Di ) by a constant ci in order to glue them together for getting an eigenfunction u. In order to go further, we first observe that this is no longer true in the case of a circle S 1 . Some compatibility condition should occur and it is rather easy to find examples for which one cannot glue together the ui ’s. As we shall see later, this phenomenon could also appear in higher dimension when Ω is not simply connected. The second observation is that this gluing procedure does not work anymore in higher dimension. In order to explain the problem, let us first give a definition. We say that two open sets Di , Dj of D are neighbors, or neighbor each other, or Di ∼ Dj if Dij := Int(Di ∪ Dj ) is connected. (1.6) Now if for two neighbors Di and Dj , λ is the groundstate energy of both H(Di ) and H(Dj ), there is no way in general to construct uij in the domain of H(Dij ) such that uij = ci ui in Di and uij = cj uj in Dj . We would indeed need at ∂Di ∩ ∂Dj the normal derivatives of ui and uj to be proportional.
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So it is natural for the analysis of the converse problem to assume in higher dimension the existence of uij for all the pairs of neighbouring domains and to try then to glue those uij so that an eigenfunction of the whole problem is obtained. We are consequently led to the following definition: Definition 1.1. We say that the partition D = {D1 , . . . , Dµ } satisfies the pair compatibility condition, for short (PCC), if, for some λ ∈ R, and for any pair (i, j) such that Di ∼ Dj , there is an eigenfunction uij 6≡ 0 of H(Dij ) such that H(Dij )uij = λuij with N (uij ) = ∂Di ∩ ∂Dj .
(1.7)
We can associate (see Figure 1 for an example) to such a partition D a graph G or G(D) by placing in each Di , i = 1, . . . , µ, a vertex vi and by associating edges eij to the vi , vj such that the corresponding Di , Dj satisfy Di ∼ Dj . We say that D is admissible if the associated graph G(D) is bipartite. Bipartite graphs are just graphs whose vertices can be colored by two colors so that vertices which are joined by an edge have different colors. This is a well known notion in graph theory, see e. g. Diestel [8]. We shall see in the next section that the nodal set N (u) of an eigenfunction u of H(Ω) has, under the condition that ∂Ω is C ∞ , the following abstract nodal set property (or shortly (ANSP)) which we now define. e (Ω) Definition 1.2 (abstract nodal set property). A closed set N in Ω belongs to N if N meets the following requirements:
(i) N is the union of smooth arcs connecting points in ∂Ω and smoothly imbedded circles in Ω. (ii) There are finitely many distinct xi ∈ Ω∩N and associated positive integers νi (νi > 2) such that, in a sufficiently small neighborhood of each of the xi , N is the union of ν(xi ) C ∞ curves (non self-crossing) two by two crossing transversally at xi (with positive angle) and such that in the complement of these points in Ω, N is locally diffeomorphic to a regular curve. (iii) ∂Ω ∩ N consists of a (possibly empty) finite set of points zj , such that, at each zj , ρj (ρj > 1) nodal lines hit the boundary. Moreover, for each zj ∈ ∂Ω, assuming that we have rotated and translated Ω such that zj = {0}, that ∂Ω is at the origin tangent to the x1 -axis and that Ω lies locally above the x1 -axis, then N is near zj the union of ρj distinct C ∞ half-curves which hit the origin with strictly positive angles.
By smooth we mean as usual that each arc, respectively circle, is a component of the zeroset of a C ∞ function which has at the zero nonvanishing gradient. The points xi will be called “critical” points of the (abstract) nodal set. Let us also observe that this definition implies that the family of abstract nodal domains which are by definition the components of Ω\ N is an admissible partition. Conversely, we can associate to a partition D the closed set [ N (D) = ∂Di \ ∂Ω, i
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5 1
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Figure 1. Partition and associated graph and will always assume that the partition is nice in the sense that [ Di . Ω \ N (D) =
(1.8)
i
Our main theorem is the following
Theorem 1.3. Suppose that Ω is simply connected with smooth boundary and that, e (Ω) and λ ∈ R, the associated family D = {D1 , . . . , Dµ } satisfies for some N ∈ N (PCC). Then there is an eigenfunction of H(Ω) with corresponding eigenvalue λ such that N (D) = N (u). Remarks 1.4. (i) The regularity assumptions can probably be relaxed, but we do not strive for generality here. See [2] for the type of conditions which could be given.
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(ii) If Ω is not simply connected then our result does not hold in general as will be explained through examples in Section 6. There we also give additional conditions on the admissible partition D such that Theorem 1.3 still holds. (iii) There are simple cases in higher dimensions for which Theorem 1.3 easily can be shown to hold (for example in some simple tree situations). This is a natural question to ask if, beyond the interest to have a theoretical criterion with minimal compatibility condition, one can check more concretely the assumptions of the main theorem. The recent contributions of Conti–Terracini– Verzini [4], [5], [6] give an interesting application1. The pair compatibility condition will appear naturally when, given an integer k and an open set Ω, one considers partitions of Ω by k open sets Di . One is then interested in the properties of optimal partitions, i. e. partitions by k open sets Di , for which maxi (λ(Di )) is minimal. A natural question is to determine when an optimal partition corresponds to a nodal pattern. As an application of our main theorem, we obtain: Corollary 1.5. Let Ω be simply connected, k ∈ N (k > 2) and let D min = (Di )i=1,...,k be a optimal admissible nice (in the sense of (1.8)) partition, such e (Ω). Then there is an eigenfunction u of H(Ω) associated with that N (D min ) ∈ N λ = max(λ(Di )), i
such that D
min
is the family of the k nodal domains of u.
Proof. Let us apply Theorem 1.3. We take as λ = maxi (λ(Di )). The first point is that all the λ(Di ) should be equal. If not, one could by deformation of the Di in a neighborhood of regular points of their boundary find a e which would decrease maxi (λ(Di )). new partition D, The second point is to observe that considering two neighbors Di and Dj , then λ should be the second eigenvalue of H(Dij ). If it was not the case for some pair (i, j), the two nodal domains of the second eigenfunction of H(Dij ) will give two new open sets Di′ and Dj′ with λ(Di′ ) = λ(Dj′ ), in contradiction with the assumption of optimality and the first point of the proof. Hence the pair compatibility condition is satisfied. Remarks 1.6. (i) The case where k = 2 corresponds to a rather well known characterization of the second eigenvalue of H(Ω). The admissibility condition is of course automatically satisfied in this case. (ii) In their sequence of papers [4], [5], [6], Conti–Terracini–Verzini, partly motivated by questions in biomathematics, have shown that there exists at least one optimal partition with these properties. P These authors consider first the minimizations of expressions of the type i (λ(Di ))p , for p ∈ (1, +∞) and the case considered here appears as a limiting (in some sense more difficult) case. (iii) In Corollary 1.5, the assumptions of regularity (and may be also the assumption that Ω is simply connected) can hopefully be removed (cf [11]), but the assumption that the partition is admissible is essential. It is easy to find examples 1We thank M. van den Berg for mentioning to us one of these articles and S. Terracini for useful discussions about these questions of optimal partitions.
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(take the disk and partitions by three open sets) where the optimal partitions do not correspond to a family of nodal domains! Organization of the paper. In Section 2 we collect some well known facts about zero sets and nodal domains. In Section 3, we analyze the properties of partitions of Ω in connection with graph theory. Section 4 is devoted to the proof of the main theorem. In Sections 5, we illustrate the theorem by discussing examples. Section 6 is devoted to a general criterion for non simply connected domains. We then analyze in Section 7 the optimality of these sufficient conditions by considering families of examples for which (PCC) does not imply a general compatibility condition. Concluding remarks are given in Section 8. 2. Regularity of Eigenfunctions and Abstract Nodal Set Property We investigate the properties of nodal domains and nodal sets. This will lead us to propose and justify the corresponding abstractions. In particular we will show that for a smooth Ω the zerosets of the eigenfunctions satisfy the abstract nodal set property introduced in Definition 1.2. First we recall some basic regularity results (cf. [9]). Proposition 2.1. Every eigenfunction u of H(Ω) belongs to C ∞ (Ω) ∩ C00 (Ω). Furthermore, for any eigenfunction u, any nodal domain ∂D is piecewise smooth and satisfies an interior cone condition. If in addition the boundary is C ∞ then u ∈ C ∞ (Ω). e (Ω). The next property justifies the introduction of N
Proposition 2.2. Suppose ∂Ω ∈ C ∞ and u is an eigenfunction of H(Ω). Then e (Ω). N (u) belongs to N Moreover the nodal lines can only cross at interior points with equal angles, and at boundary points crossing nodal half-lines determine also together with the boundary equal angles. The proof follows rather directly from the local behaviour of eigenfunctions near their zeros. Lemma 2.3. Let u in H01 (Ω∩B(x0 , ρ0 )) with Ω∩B(x0 , ρ0 ) 6= ∅ where B(x0 , ρ0 ) = {x ∈ R2 : |x − x0 | < ρ0 } and x0 ∈ Ω. Let us assume that: (−∆ + V − λ)u = 0
in Ω ∩ B(x0 , ρ0 ).
(a) Suppose x0 ∈ N (u) ∩ Ω then there is ν such that, u(x) = Pν (x − x0 ) + Pν+1 (x − x0 ) + O(|x − x0 |ν+2 ),
(2.1)
in a sufficiently small neighborhood of x0 . Thereby Pν 6≡ 0, defined by Pν (x) = r ν (a cos νω + b sin νω),
(2.2)
is a harmonic homogeneous polynomial of degree ν. For simplicity we have written this in polar coordinates r, ω. Note that the zeroset of any Pm consists of m straight lines which intersect with equal angles.
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(b) If z0 ∈ N (u)∩∂Ω then there exist ρ > 1 and r0 > 0 such that, in Ω∩B(z0 , r0 ), u(x) = Pρ+1 (x − z0 ) + O(|x − z0 |ρ+2 ).
(2.3)
Thereby Pρ+1 6≡ 0 in (2.3) is defined as in (2.2) and has the property that the line tangent to ∂Ω at z0 is in the zeroset of Pρ+1 (x − z0 ). About the proof of Lemma 2.3. This lemma is well known among specialists (see for instance [3] or [12]). A detailed proof of the boundary case is for example given in [10] using the reflection argument along the lines of [13]. Remarks 2.4. (i) Note that for each component of ∂Ω the number of nodal lines hitting this component has to be even. This is implied by the property that the graph associated to N is bipartite. Indeed, otherwise the associated graph would contain an “odd circle” (that is a circle with an odd number of vertices) and this would make it impossible to color the associated graph with two colors. (ii) It is useful to have also a description of the zeros of eigenfunctions in the case of domains with piecewise C ∞ boundaries and to describe the local structure of the zeros near the corners. This is discussed for example in [7]. The main difficulty for having a local structure lemma at the corner is to show that the solution cannot be flat at the corner, that is cannot decay faster than polynomially. This can be proved2 (see [1] or [11]) by the use of a conformal transformation. 3. Graphs and Circulation Along Paths 3.1. Preliminaries. We start with a partition D = {D1 , . . . , Dµ } so that [ e (Ω). N= ∂Di \ ∂Ω ∈ N
We assume that for some given λ, D satisfies (PCC). We introduce various normalizations. For i = 1, . . . , µ, let ui be the positive normalized groundstate of H(Di ). Similarly, for any oriented pair (i, j) such that Di and Dj are neighbors, let uij be the eigenfunction of H(Dij ) having ∂Di ∩ ∂Dj as nodal set, hence Di and Dj as nodal domains. We normalize uij and impose that uij is strictly positive in Di . Then uij is uniquely determined and negative in Dj . Note that with this choice uij = −uji , (3.1) and that we can write uij = dij (ui − γij uj ),
(3.2)
with γij and dij strictly positive. We can then write γij = exp cij , and we call cij the circulation from Di to Dj . Having in mind (3.1), we get the relations : dij = dji γji ,
dij γij = dji .
2We thank S. Terracini for explaining to us the proof.
(3.3)
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In particular we get the important relation: γij γji = 1,
(3.4)
which in the circulation terminology becomes cij + cji = 0.
(3.5)
3.2. Good paths. We now consider continuous paths in Ω wandering between the nodal domains. More precisely the following notion of good path is useful: Definition 3.1. We say that the path [0, 1] ∋ t 7→ β(t) ∈ Ω is good (with respect to D) if (i) β(0), β(1) ∈ Ω \ N , β(t) ∩ ∂Ω = ∅ for t ∈ [0, 1]. (ii) β(t) ∩ N (ν) = ∅ for ν > 2 where N (ν) = {x ∈ N ∩ Ω : ν nodal lines locally cross in x}. (iii) If for some t0 ∈ (0, 1), β(t0 ) ∈ ∂Di ∩ ∂Dj , then there is an ǫ > 0 such that, for t ∈ (t0 − ǫ, t0 ), β(t) ∈ Di (or Dj ) and for t0 ∈ (t0 , t0 + ǫ), β(t) ∈ Dj (or Di ). To any good path, we can associate a finite sequence i0 , i1 , . . . , ik of indices expressing the restriction of the path to the graph G. We call this restriction the associated G-path and denote it by βG . This simply means that the path starts from β(0) ∈ Di0 , then leaves Di0 for entering in Di1 and a new index is added at each crossing of a boundary. The length of the path is then exactly the number of crossings of the path. As usual, we say that the path is closed if β(0) = β(1). Note that if β(0) and β(1) belongs to the same Di then we can always close the path (using the property that Di is arcwise connected), keeping the corresponding graph fixed, which is in any case a circle. 3.3. Circulation along βG . We can associate to each good path β(t) two numbers γβ =
k−1 Y
γiℓ ,iℓ+1 ,
k−1 X
ciℓ ,iℓ+1 ,
ℓ=0
and Cβ =
ℓ=0
the second one being called the circulation along β. Of course, we have γ β = exp Cβ . Note that Cβ depends only on the G-path βG , so Cβ will also be called the circulation along βG . When we deform these paths by homotopy, it is clear that as long as the path keeps the property of being good the circulation is constant. But one of our goals will be to follow this circulation when changing in the homotopy the corresponding G-path (that is the path in the homotopy does not remain a good path).
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4. Proof of the Main Theorem 4.1. General compatibility condition. The general condition for constructing an eigenfunction of H(Ω) by gluing the uij of H(Dij ) is quite reminiscent of the problems occuring in Cech Cohomology3 but we have finally preferred to remain selfcontained, instead of sending the reader to a too sophisticated book in cohomology. The following criterion4 is quite natural: Proposition 4.1. If we have an admissible partition D in Ω generated by some N e (Ω) such that (PCC) holds for some λ ∈ R, then an eigenfunction of H(Ω) in N associated with λ can be constructed if and only if, for any closed path on the graph of length k > 2, the condition (GCC): k−1 Y
γiℓ ,iℓ+1 = 1,
(4.1)
ℓ=0
is satisfied.
In other words, the circulation along any closed path on the graph must be 0. Remarks 4.2. (i) At this stage, it is not necessary to assume that Ω is simply connected. (ii) Note that because we are in a bipartite graph, k has to be even in (4.1). Proof of Proposition 4.1. One starts from one domain Di0 and from its groundstate ui0 . Then the extension of ui0 to all neighboring domains is obtained by using assumption (PCC). One can then propagate the extension to the next neighbors till Ω is covered. (GCC) just permits a construction which is independent of the path used for the extension. This gives a global construction of an element u in 2 H01 (Ω) belonging to Hloc (Ω \ N c ) where N c is the set of critical points of N lying in Ω, and satisfying in the distribution sense (−∆ + V − λ)u = 0 in Ω \ N c . But (−∆ + V − λ)u belongs to H−1 (Ω) so it remains to show that a distribution in H−1 with support in N c is 0. But N c is a finite set and it is enough to recall the standard result in distribution theory: Lemma 4.3. If ω is an open set in R2 and x0 ∈ ω, there are no distribution in −1 Hloc (ω) with support in {x0 }. This achieves the proof of the proposition.
3Note however that we are not working with coverings but with partitions and that the γ ij are only defined for pairs of neighbors in the specific sense given in the introduction. So the γij ’s define a cochain on Ω′ corresponding to Ω \ ∪i xi , where the xi ’s denote the singular points of the abstract nodal set. 4 In the algebraic topology language, the criterion says essentially that, when our cochain is a cocycle, it is a coboundary.
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So the proof of our main theorem consists in showing that Condition (4.1) is always satisfied when Ω is simply connected. In particular, it is not too difficult to see that the condition is automatically satisfied in the case when the graph associated to the partition is a tree. 4.2. Further reduction. We now explain how to reduce the computation of the circulation along a good closed path to the case where a given good path encloses only one or no critical points. The proof is by induction. Suppose that a closed good path γ parametrized by t ∈ [0, 1] encloses exactly k critical points (with k > 2). The claim is that we can find two points (inside nodal domains) on this closed path corresponding to times t0 and t1 , and construct a continuous curve ℓ01 going from γ(t0 ) to γ(t1 ) avoiding the critical set such that the path γ1 defined by γ1 = γ on [0, t0 ], by ℓ01 on [t0 , t1 ] (after reparametrization) and by γ1 = γ on [t1 , 1] is a good path containing only in its interior one critical point. If now ℓ10 denotes the opposite path to ℓ01 , we can consider the closed good path γ2 such that γ2 = ℓ10 on [0, t0 ], γ2 = γ on [t0 , t1 ]. It is clear that γ2 encloses in its interior (k − 1) critical points and that the circulation along γ is the sum of the circulation along γ1 and of the circulation along γ2 . If (k − 1) > 2, we can iterate the procedure till each path encloses at most one critical point. So the general proof is reduced to the analysis of the condition (4.1) in the case when a good closed path either encloses no critical point or one critical point. This will be the object of the two next subsections. We will actually show that in these two situations the circulation along the path is zero, when the path is homotopic in Ω to a point, which is automatically the case if Ω is simply connected. 4.3. Proof when a good closed path does not enclose any critical point and is homotopic to a point. In this case we can find a homotopy γ(s, t) such that γ(0, t) is the initial path, γ(1, t) is a single point living in some Di . Of course the graph trace of the path, i. e. βG , is changing with this homotopy. But since each nodal domain has only finitely many critical points in its boundary a continuity argument shows that, by modifying the homotopy, we can get one for which there are only finitely many sk for which the paths are no more good paths. Moreover one can pick this homotopy in such a way that, at these sk , the paths γ(sk , ·) have still the property that they are good except at one point tk . So the transition near the point (denoted by S0 in the figure) γ(sk , tk ) is the following (or the converse). There exists some pair of neighboring Di and Dj such that γ(sk , tk ) ∈ ∂Di ∩ ∂Dj . For s < sk , with (s, t) near (sk , tk ), the path γ(s, t) (see Figure 2) is contained in Diℓ , with iℓ = i. For s = sk and t near tk , γ(sk , t) belongs to Diℓ except at γ(sk , tk ). For s > sk with s near sk , the path enters the neighbor Dj = Dinew , before returning to Diℓ and entering Diℓ+1 . In particular we can pick ℓ+1 the homotopy always so that the path avoids any critical point. So the initial corresponding G-path i1 , i2 , i3 , . . . , iℓ−1 , iℓ , iℓ+1 , . . . , ik becomes i1 , i2 , i3 , . . . , iℓ−1 , iℓ , inew ℓ+1 , iℓ , iℓ+1 , . . . , ik . The fact that the circulation is conserved in this transformation is an immediate consequence of (3.4).
CONVERSE SPECTRAL PROBLEMS FOR NODAL DOMAINS
D il+1 Di l
D il+1
D il+1 S0
77
D
Dinew l+1
new il+1
S0
Di l
Di l
S0
Dinew l+1
Figure 2. Deformation argument in three pictures: before, at the touching and after
For the converse transition, we just replace a sequence iℓ , inew ℓ+1 , iℓ , iℓ+1 by iℓ . After finitely many operations of this type, we will obtain a path reduced to a single point whose G-path is also a point. Actually the main point here is that the associated closed G-path is a path on a tree! 4.4. Proof when a good closed path encloses a unique critical point and is homotopic to a point. Let us now consider the case of a closed path encloses a unique critical point. We can reduce the computation to the case when this closed good path is a small circle turning once and positively around this point x0 . There exists ν > 2 so that the ν nodal arcs σ0 , σ1 , . . . , σν−1 pass locally through x0 . This means that there is an ǫ0 > 0 such that for 0 < ǫ < ǫ0 N ∩ B(x0 , ǫ) =
ν−1 [
σj ∩ B(x0 , ǫ)
j=0
where B(x0 , ǫ) = {x ∈ R2 : |x − x0 | < ǫ}. We shall also use the (2ν) half arcs σℓ+ + (such that σℓ = σℓ+ ∪ σℓ+ν ). Without loss we might assume that x0 = {0} and that σ0 is tangent to the x1 -axis at x0 . The arc σ0 splits B(x0 , ǫ) in two parts and we denote by B + (x0 , ǫ) the upper part which lies “above” σ0 . σ1
σ2
σ1
+
+ σ2
S3 σ3+
+
B(x0,ε)
S2
x0
S6
S4 S5 σ4+
σ0 + + σ0 = σ6
S1
σ5+
Figure 3. Picture in the case when ν = 3
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Assume now (PCC) for D. The abstract nodal set (intersected with B(x0 , ǫ)) defines 2ν curved sectors Sℓ (also intersected with B(x0 , ǫ))5 (ℓ = 1, . . . , 2ν) de+ limited6 by σℓ−1 and σℓ+ , each one belonging to some Diℓ . Take the first domain S1 starting from σ0+ . Starting from ui1 , the pair compatibility condition can be used iteratively to extend the restriction of ui1 to S1 as a local solution v1 ∈ C ∞ (B + (x0 , ǫ)) such that (−∆ + V )v1 = λv1 in B + (x0 , ǫ), v1 = 0 for N ∩ B + (x0 , ǫ) and v1 = ui1 on S1 . We can now apply Lemma 2.3 (b) to v1 in B + so that7 v1 = c1 r ν sin νω + O(r ν+1 ) for some constant c1 6= 0. In particular this means that though we have not assumed that the nodal lines cross at the point x0 under equal angles this is enforced by Lemma 2.3. The second point is that by restriction to S1 , we get ui1 = c1 r ν sin νω + O(r ν+1 ) in S1 . and a similar expansion is true for ui1 i2 : ui1 i2 = c12 r ν sin νω + O(r ν+1 )
in Int(S 1 ∪ S 2 ).
Of course we can do the same thing starting from any Sℓ . So, for each ℓ, we have shown the existence of cℓ > 0 such that uiℓ has the asymptotics uiℓ = −cℓ r ν sin (ν(ω − ωℓ,ℓ+1 )) + O(r ν+1 ),
(4.2)
in Sℓ , where ωℓ,ℓ+1 is the argument of the tangent to ∂Sℓ ∩ ∂Sℓ+1 at 0 and a similar expansion holds for uiℓ iℓ+1 . But reusing (PCC) (through (3.2)) gives that cℓ+1 . (4.3) γiℓ ,iℓ+1 = cℓ Coming back to the definition of the circulation and using (4.3) we get the vanishing of the circulation along the good path enclosing the critical point. This completes the consideration of this case and finishes also the proof of the main theorem. 5. Examples We apply the general constructions above for the analysis of specific examples. 5.1. Three examples whose corresponding graph is a tree A simple partition. The left subfigure in Figure 4 presents a partition by five domains, whose corresponding graph is a tree. Moreover, there are no critical points inside the domain. 5We omit from now on recalling the fact that we are always in a small ball around x . But 0
the whole proof is local. See Figure 3 for the picture in the case ν = 3. 6By convention, σ + is σ + . 2ν 0 7Here there are two possibilities in the choice of polar coordinates. We can either flatten the boundary of B + and take polar coordinates after flattening or keep the initial ones. In any case, the two choices lead to the same main term.
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The eight. Take the case of the eight as in the central subfigure of Figure 4 with (0, 0) as critical point. Let D2 and D3 the two interior “abstract” nodal domains and D1 the “exterior” nodal domain meeting ∂Ω. Then D1 ∼ D2 and D1 ∼ D3 . Suppose that, for some λ, we have (PCC). We apply Lemma 2.3 (case (b)) locally r0 ,± r,± to u12 in D12 , (r0 > 0 small enough), where D12 = Int(D2r ∪ D1r,± ), D1r,± being the two components (for r small enough) of D1 ∩ B(0, r). This shows that the groundstate u1 (which is up to a multiplicative constant the restriction to D1 of u12 ) satisfies u1 = c1 r 2 sin(2(ω − ω± )) + O(r 3 ), near x0 = 0, where the constant c1 is the same for the two “opposite” sectors describing D1 near x0 . But the associate graph is a tree. It is trivial in this case that the trace of a good closed path on the graph has always zero circulation. We do not need to use the information given by the local analysis around the critical point. The only additional information given by this analysis is that u1 has the same asymptotics near 0 in the two opposite sectors. The clover leaf intersection. This example (right subfigure in Figure 4) does not lead to any difficulty. The graph is a tree. On can directly extend from D1 toward respectively D2 , D3 and D4 .
1 4
2
3
5
1
1 2
3
3 4
2
1 5
4
1
1
3 2
2
3
2
3
4
Figure 4. Three examples with associated graph below: simple, eight and clover
5.2. Examples with circles The cross. When Ω is the disk B(0, 1), the cross (say {x1 = 0} ∪ {x2 = 0} determines four nodal domains Dj (j = 1, 2, 3, 4) (see the left subfigure in Figure 5 and its corresponding graph below) so we have a “circle” (1, 2, 3, 4). The corresponding graph can be represented by a square. Here we cannot avoid the local analysis around the center.
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1 1
4
4 3
2
2
3
1
4
1
4
2
3
2
3
Figure 5. Two other examples in simply connected domains with square graph Two intersecting circles. Although the corresponding graph is the same, the nodal structure is different. There are two critical points. There are three typical paths enclosing critical points. One is living in D1 and turning positively and once around D2 ∪D3 ∪D4 . By definition, the circulation along this path is trivial. The second one is a small positively oriented circle around the left critical point. Its corresponding trace in the graph is the sequence (1, 2, 3, 4). Here we need to perform the local analysis. The third one is a small positively oriented circle around the right critical point. Its corresponding trace in the graph is the sequence (4, 3, 2, 1). One can perform the local analysis but also observe that the circulation along this path is just the opposite of the previous one. 6. Sufficient Conditions in the Non Simply Connected Case In the non simply connected case, what remains from the previous proof can be formulated as follows. Proposition 6.1. In each homotopy class of Ω, all the good paths have the same circulation. In particular we can speak of a circulation attached to a homotopy class. Remark 6.2. As a consequence of Proposition 6.1, for a given partition satisfying (ANSP) and (PCC), the proof that (GCC) holds is reduced to the proof that in each homotopy class there is a representative with circulation 0. Of course we recover in the simply connected case Theorem 1.3. Remark 6.3. It is actually enough (using the properties of the fundamental group of Ω) to verify (GCC) for a set of generators of this group.
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In this spirit the case of one hole can be treated in greater detail. The homotopy group is generated by the (class of) simple path (s) turning once and anticlockwise around the hole. This leads to the following sufficient condition: Proposition 6.4. We assume that Ω has just one hole and that, for some partition D and λ, (PCC) is satisfied. If there is a good path of index 1 around the hole, which intersects the abstract nodal set at at most two points, then the conclusions of Theorem 1.3 hold. The proof is immediate using Proposition 6.1, Remark 6.2 and the fact that the circulation along a good path with no crossing is 0 by definition and is also 0 in the case of two crossings by (PCC). Typically the assumptions are satisfied when the hole or the exterior boundary are hit by no or two nodal lines. 7. On the Optimality in the Non Simply Connected Case Let us consider the case with one hole and let λ be an eigenvalue of multiplicity 1 such that the corresponding eigenfunction u has at least four nodal domains. We would like to present a family of examples for which one can then construct a new potential so that the main theorem does not hold any more in spite of (PCC). Assumption 7.1. We assume that there exists one nodal domain D such that Ω \ D becomes simply connected and such that the boundary of D with each of its neighbors is connected. We also assume that ∂D ∩ ∂Ω meets the regular parts of the exterior boundary and of the interior boundary. Then the claim is Proposition 7.2. Under the previous assumptions, we can find a new potential Vǫ such that λ satisfies the pair compatibility condition corresponding to N (u) and such that λ is not an eigenvalue of −∆ + Vǫ with an eigenfunction having the same nodal domain as u. The proof is inspired by the analysis of the case of the circle presented in the introduction. By assumption ∂D contains two distinct non crossing continuous curves L± joining the two boundaries. We proceed by constructing a C ∞ function b in D, such that the support of ∇b does not meet L± , b = 0 near L+ , and b = b− near L− , where b− ∈ R \ {0}. Moreover, we can require that ∇b · n = 0 on ∂Ω ∩ ∂D,
(7.1)
where n is the outward normal to ∂Ω. We note that b can be extended by 0 outside D to Ω \ L− but not to Ω. On the contrary, ∇b and ∆b can be extended to the whole Ω! We now introduce uǫ = (1 + ǫb)u. (7.2)
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L-
b=bb=0 L+ supp ∇b
Figure 6. Construction of b in the simplest case We note that uǫ is well defined in Ω \ L− and can also admit an extension, when crossing L− , by (1 + ǫb− )u to the neighbors of D touching D in L− . Moreover, if ǫ 6= 0, it cannot be extended to a C ∞ function in Ω. We now claim that, for small enough ǫ 6= 0, uǫ is an eigenfunction of −∆ + Vǫ in Ω \ L− , with eigenvalue λ and with ∆b ∇b · ∇u Vǫ := V + ǫ +2 . (7.3) 1 + ǫb uǫ We observe, using the property (7.1) and that ∇u is not vanishing on the boundary on the support of ∇b, by Hopf’s boundary point Lemma (see [9]), that Vǫ admits a C ∞ extension to Ω. Now (PCC) is satisfied for λ, −∆ + Vǫ in Ω and the family associated to N (u). If λ was an eigenvalue of the Dirichlet realization of −∆+Vǫ in Ω with a corresponding eigenfunction vǫ with nodal set N (u), then comparing vǫ and uǫ in D, we would get vǫ = cǫ uǫ . Remark 7.3. Note that we do not know if λ is an eigenvalue of the Dirichlet realization of −∆ + Vǫ in Ω. But vǫ is C ∞ in Ω and uǫ has a discontinuity! Hence a contradiction. Remarks 7.4. (i) More general families of examples are treated in [10]. (ii) One could ask naturally if replacing (PCC) by the triple compatibility condition (TCC) will lead to other results. Using the same ideas as above, it is easy to construct examples for which (TCC) does not imply (GCC). 8. Final Remarks In this paper we have analyzed some of the properties (local, global, spectral) satisfied by a family of sets formed by nodal domains of an eigenfunction.
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We have then proposed a sufficient natural pair compatibility condition permitting to glue together eigenfunctions attached to each pair of neighboring domains. We have shown its sufficiency in the case when Ω is a simply connected open set in R2 and described how one can extend the analysis in the non simply connected situation. The analysis of a family of examples shows that the sufficient conditions we have proposed are in some sense not far from optimal. Except trivial cases, where no circle in the corresponding graph can occur, the analysis of the same question in dimension > 2 is completely open. A precise description of the structure of the nodal set of the eigenfunction near its critical points is indeed missing. Let us finally mention that the same problem can be considered for a Schr¨ odinger operator on a surface either with or without boundary. Then the genus has to e (Ω) play a role. While for the flat case and actually also for the sphere N ∈ N automatically guarantees that the nodal domains created by Ω \ N lead to an admissible family D this has to be required for the case of surfaces in general. Take for instance the torus: then an N which is just a simple closed loop which is not zero-homotopic creates just one nodal domain, hence not an admissible D. Acknowledgements. We thank S. Alinhac, M. Dauge, S. Fournais, H. Koch, L. Robbiano and T. Ramond for help or discussions around Strong Uniqueness theorems and various aspects of the theory of boundary value problems in domains with corners. We thank also M. van den Berg and S. Terracini for instructive discussions about the question of optimal partition. References [1] G. Alessandrini, Nodal lines of eigenfunctions of the fixed membrane problem in general convex domains, Comment. Math. Helv. 69 (1994), no. 1, 142–154. MR 1259610 [2] A. Ancona, B. Helffer, and T. Hoffmann-Ostenhof, Nodal domain theorems ` a la Courant, Doc. Math. 9 (2004), 283–299 (electronic). MR 2117417 [3] L. Bers, Local behavior of solutions of general linear elliptic equations, Comm. Pure Appl. Math. 8 (1955), 473–496. MR 0075416 [4] M. Conti, S. Terracini, and G. Verzini, An optimal partition problem related to nonlinear eigenvalues, J. Funct. Anal. 198 (2003), no. 1, 160–196. MR 1962357 [5] M. Conti, S. Terracini, and G. Verzini, On a class of optimal partition problems related to the Fuˇ c´ık spectrum and to the monotonicity formulae, Calc. Var. Partial Differential Equations 22 (2005), no. 1, 45–72. MR 2105968 [6] M. Conti, S. Terracini, and G. Verzini, A variational problem for the spatial segregation of reaction-diffusion systems, Indiana Univ. Math. J. 54 (2005), no. 3, 779–815. MR 2151234 [7] M. Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. MR 961439 [8] R. Diestel, Graph theory, Graduate Texts in Mathematics, vol. 173, Springer-Verlag, New York, 2000. MR 1743598 [9] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190 [10] B. Helffer and M. Hoffmann-Ostenhof, Converse spectral problems for nodal domains, Extended version. Preprint, 2005. [11] B. Helffer, M. Hoffmann-Ostenhof, and S. Terracini, work in progress. [12] T. Hoffmann-Ostenhof, P. W. Michor, and N. Nadirashvili, Bounds on the multiplicity of eigenvalues for fixed membranes, Geom. Funct. Anal. 9 (1999), no. 6, 1169–1188. MR 1736932
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[13] L. Robbiano, Fonction de coˆ ut et contrˆ ole des solutions des ´ equations hyperboliques, Asymptotic Anal. 10 (1995), no. 2, 95–115. MR 1324385 D´ epartement de Math´ ematiques, Bat. 425, Universit´ e Paris-Sud, 91 405 Orsay Cedex, France E-mail address: [email protected] ¨r Theoretische Chemie, Universita ¨t Wien, Wa ¨hringer Strasse 17, A-1090 Institut fu Wien, Austria, and ¨ dinger Institute for Mathematical Physics, BoltzmanInternational Erwin Schro ngasse 9, A-1090 Wien, Austria E-mail address: [email protected]
