Minimal partitions for anisotropic tori
MMinimal partitions for anisotropic tori
B. Helffer (Universit´e Paris-Sud 11)andand T. Hoffmann-Ostenhof (University of Vienna)
Abstract
We analyze spectral minimal k -partitions for the torus. In contin-uation with what we have obtained for thin annuli or thin strips on acylinder (Neumann case), we get similar results for anisotropic tori. Key words : spectral theory, minimal partition, Laplacian, nodalsets
For a ≥ b >
0, consider the Laplacian on the 2 D -torus : T ( a, b ) := S ( a π ) × S ( b π ) . Concretely, we can also consider R ( a, b ) = (0 , a ) × (0 , b ) , (1.1)and the Laplacian on R ( a, b ) with periodic boundary conditions but exceptfor the pictures this is not the most convenient point of view. It is indeedbetter to think of the torus as a compact regular manifold.We can, following [5], consider k -partitions D of the torus, i.e. families ofdisjoint open sets ( D , . . . , D k ) of the torus and the sequence of partitionenergies L k ( T ( a, b )) obtained by minimizing over D of the torus some energydefined by Λ k ( D ) = max j λ ( D j ) , (1.2)where λ ( D j ) is the ground state energy of the Dirichlet Laplacian in D j . Wethen define L k ( T ( a, b )) := inf D Λ k ( D ) , (1.3)1 a r X i v : . [ m a t h . SP ] S e p here the infimum is over all the k -partitions of T ( a, b ). A minimal k -partition is a partition whose energy is L k ( T ( a, b )) . As in the case of anopen set in R , minimal k -partitions exist and are strong and regular (seeSection 2). Without loss of generality, we consider the case a = 1. Note thatfor the torus, when b < λ = 0 and that λ = λ = 4 π . Hence L > λ and using the results of [5] (extended to the case of the torus) (see Theorem2.1 in Section 2) the associated minimal 3-partition cannot be nodal, i.e apartition obtained as the nodal domains of an eigenfunction. On the otherhand for k = 4, we see that λ = 16 π for b < and that any correspondingeigenfunction has four nodal domains. So the minimal 4-partition is nodal.Our aim in this paper is to describe what are the minimal k -partitions. Ourmain result is the following: Theorem 1.1
There exists b k > such that, if b < b k , L k ( T (1 , b )) = k π and the cor-responding minimal k -partition D k = ( D , . . . , D k ) is represented in R (1 , b ) by D i = ( ( i − /k , i/k ) × [0 , b ) , for i = 1 , . . . , k . (1.4) Moreover we can take b k = k for k even and b k = k for k odd. Note that the boundaries of the D i in T (1 , b ) are just k circles (see Figure 1where these circles are represented by vertical segments).Figure 1: One candidate for the minimal 3-partition represented in R (1 , b ). emark 1.2 This result is a complement to what we have obtained for thin annuli or stripson a cylinder (in the case of the Neumann condition) [4]. Its proof requiresnew ideas which hopefully can be used for other compact surfaces. We recallthat the case of thin annuli with Dirichlet conditions is still open ( k odd).For minimal k -partitions of the torus, we will at the end of Section 2 provethat the statement of the theorem holds for k even (the minimal partitionsare nodal) and b k = k cannot be improved (see also Section 7 for furtherdiscussion). For k even and k < b < k − , the k -th eigenfunction does nothave k nodal domains. Hence it remains to give the proof of our theorem for k odd ( k ≥ ).We also recall that in the case k = 3 , the problem was solved in [6] for thesphere S and is still open for the disk [6] and the square [1]. Let us first recall in more detail the properties of minimal k -partitions. Thenotion of minimal partition was first introduced for an open set Ω in R in[5] (see references therein). We just present the corresponding definitionsfor the torus (or more generally on a compact Riemannian manifold). Werecall that a k -partition on the torus is simply a family D of k -disjoint opensets ( D i ) i =1 ,...,k . Such a partition is called strong if ∪ D i = T (1 , b ) andInt ( D i ) = D i for any i . Attached to a strong partition, we associate a closedset in T (1 , b ), which is called the boundary set of the partition : N ( D ) = ∪ i ∂D i . (2.1) N ( D ) plays the role of the nodal set (in the case of a nodal partition). Wehave recalled in the introduction the notion of minimal k -partitions. As inthe case of an open set in R , minimal k -partitions exist and are strong and regular in the following sense. We call a partition D regular if its associatedboundary set N ( D ), has the following properties :(i) Except for finitely many distinct x i ∈ N in the neighborhood of which N is the union of ν i = ν ( x i ) smooth curves ( ν i ≥
3) with one end at x i , N islocally diffeomorphic to a regular curve.(ii) N has the equal angle meeting property . The x i are called the crit-ical points and define the set X ( N ). By equal angle meeting property ,we mean that the half curves meet with equal angle at each critical point of3 .In the case of an open set we have also points y j at the boundary and we callthis set Y ( N ).Moreover, the minimal k -partitions are bipartite, i.e. can be colored bytwo colors (neighboring domains have different colors), if and only if they arenodal (i.e. corresponding to the nodal domains of an eigenfunction of theLaplace-Beltrami operator). Another important statement established in [5]is: Theorem 2.1 A k -partition consisting of the k nodal domains of an eigenfunction corre-sponding to the k -th eigenvalue λ k of the Laplacian is a minimal k -partition. In general one could just say that by the well known Courant nodal theoremthe number of nodal domains of an eigenfunction u k associated with λ k is atmost k . The eigenpair ( u k , λ k ) is called Courant sharp if the number ofnodal domains is exactly k . Theorem 2.1 is moreover optimal as has beenproven in [5]: Theorem 2.2
A nodal minimal k -partition corresponds necessarily to a Courant sharp pair. First application: proof of Theorem 1.1 in the even case.
For the torus T ( c, d ), the eigenvalues are given by 4 π ( m c + n d ) (( m, n ) ∈ N where N denotes the set of the non-negative integers) with a correspondingbasis given by • ( x, y ) (cid:55)→ cos(2 πm xc ) cos(2 πn yd ), • ( x, y ) (cid:55)→ cos(2 πm xc ) sin(2 πn yd ), • ( x, y ) (cid:55)→ sin(2 πm xc ) cos(2 πn yd ) • and ( x, y ) (cid:55)→ sin(2 πm xc ) sin(2 πn yd )(with suitable changes when m or n vanishes). For example, for n = 0, weget ( x, y ) (cid:55)→ m = 0 and ( x, y ) (cid:55)→ cos(2 πm xc ) and ( x, y ) (cid:55)→ sin(2 πm xc )for m >
0. These eigenfunctions have (2 m ) nodal domains on the torus.When k is even and k < cd , we get the existence of an k -th eigenfunctionwith exactly k nodal domains (corresponding to m = k and n = 0). What4ill be important in our problem is that Theorem 2.1 implies that for k evenand c > d > k -partitions of T ( c, d )) are nodal for the case that k < c/d . The corresponding energy is π k c . Hence we have completed theproof of Theorem 1.1 for the even case.We also observe that for k odd ( k >
1) the minimal k -partitions cannotbe nodal.We will prove that, when cd is small enough, the minimal k -partitions can belifted into a Courant sharp (2 k )-partition on the covering T (2 c, d ).The k -partition appearing in Theorem 1.1 corresponds actually to a nodal partitionon this covering and this implies the result. The existence of this lifting willbe proved in Section 5. The computation of the energy of the k -partition (1.4) leads immediately tothe following upper bound for L k . Proposition 3.1 k π min(1 , b − ) ≥ L k ( T (1 , b )) . (3.1)Using this upper-bound we can give necessary conditions on k -partitionsto be minimal. Proposition 3.2 If b < k , there is no minimal k -partition D = ( D , . . . , D k ) of the torus withone D i homeomorphic to a disk. The proof is by contradiction. Let D = ( D , . . . , D k ) be a minimal k -partition such that, say D is homeomorphic to a disk. Then, the pullback (cid:98) D of D in the universal covering R is a union of bounded components (cid:98) D k,(cid:96) (with ( k, (cid:96) ) ∈ Z ) such that (cid:98) D k,(cid:96) + ( m, nb ) = (cid:98) D k + m,(cid:96) + n . Moreover (cid:98) D , hassame area as D and λ ( D ) = λ ( (cid:98) D , ) .Looking at a lower bound for λ ( (cid:98) D , ), one could first think of using Faber-Krahn’s inequality but it is better to come back to the first step of one proofof the Faber-Krahn inequality which is based on the Steiner symmetrization(see for example the book [7] (Section 2.2) or the expository talk [9]).We now observe that each vertical slice has a total length less than b . We5ow apply the Steiner symmetrization with respect to the horizontal line y = b . It is immediate to see that the image S ( (cid:98) D , ) of (cid:98) D , is contained ina rectangle (cid:98) R b in the form ( − (cid:96) b , (cid:96) b ) × (0 , b ) for some (cid:96) b >
0. Now it is wellknown that in this symmetrization we have: λ ( (cid:98) D , ) ≥ λ ( S ( (cid:98) D , )) , and by monotonicity λ ( S ( (cid:98) D , )) ≥ λ ( (cid:98) R b ) = π ( b − + (cid:96) − b ) > π b − . This leads to λ ( D ) > π b − , (3.2)hence, using (3.1), to b > k . (3.3)This gives the contradiction. In the case of an open set Ω in R , observing that the Euler characteristic ofΩ is 2, we have for a regular minimal k -partition D : k = b − b + 1 + (cid:88) i (cid:18) ν ( x i )2 − (cid:19) + 12 (cid:88) j ρ ( y j ) . (4.1)where b is the number of components of ∂ Ω, b is the number of componentsof ∂ Ω ∪ N , ν ( x i ) and ρ ( y j ) the numbers of arcs associated with the singularpoints x i ∈ X ( N ) of the boundary set N = N ( D ) in Ω, respectively with thepoints y j of the boundary set contained in ∂ Ω. We denote by X ( N ) the setof the x i ’s and by Y ( N ) the set of the y j ’s. In the case of a flat compact surface M without boundary, it is easier toformulate Euler’s formula by using the Euler’s characteristics of M and of6he elements of the partition D = ( D , . . . , D k ). The formula reads (cid:88) (cid:96) χ ( D (cid:96) ) = χ ( M ) + (cid:88) i (cid:18) ν ( x i )2 − (cid:19) , (4.2)and is a direct consequence of the Gauss-Bonnet formula applied in each D i (see for example [8]).We recall that for the torus: χ ( T ( a, b )) = 0, for the disk B : χ ( B ) = 1, forthe annulus A : χ ( A ) = 0 and for the sphere S : χ ( S ) = 2. Hence, in thecase of the torus, (4.2) becomes: k (cid:88) (cid:96) =1 χ ( D (cid:96) ) = (cid:88) i (cid:18) ν ( x i )2 − (cid:19) . (4.3) Proposition 4.1
A minimal partition D = ( D , ..., D k ) for which no D (cid:96) is homeomorphic tothe disk satisfies X ( N ( D )) = ∅ . Proof.
The assumption implies that χ ( D (cid:96) ) ≤
0, for (cid:96) = 1 , . . . , k . Then we immedi-ately get from (4.3) that χ ( D (cid:96) ) = 0 and that X ( N ) = ∅ . Proposition 5.1
Suppose D = ( D , . . . , D k ) is a minimal k-partition on the torus T (1 , b ) forwhich all the D i are not homeomorphic to the disk and X ( N ( D )) = ∅ . Then D can be lifted to a bipartite (2 k ) - partition of T (2 , b ) . The initial guess was that a double covering will suffice but this is notalways the case. One can construct (see Figure 2) a 3-partition of the toruswithout critical point, for which it is necessary to construct a covering oforder 4, T (2 , b ) of the torus (doubling in each direction) in order to get abipartite 6-partition (see Figure 3). Proof of Proposition 5.1.
One can classify all the possible topological types of these partitions. The k open sets of the partition have the same topological type. Each open set can Thanks to P. B´erard for giving us the reference.
7e deformed by a retraction onto a simple closed line without self-intersection.Hence the classification corresponds to the classification of closed lines onthe torus without self-intersection that are not homotopic to a point (theso-called torus knots). They correspond (see [2], p. 47, Example 1.24) tolines generically denoted by (cid:96) p,q turning p times around one horizontal circleand q times around the vertical one, with p and q mutually prime (exceptif q = 0 , p = 1 or p = 1, q = 0). Figure 2 corresponds to p = 1, q = 1.The candidate for the minimal 3-partition when b is small corresponds to p = 1, q = 0. Another example is given in the first subfigure of Figure 4,which represents a closed line on the torus with p = 3 and q = 2. We goto a suitable double covering so that either p or q is multiplied by 2; so thegreatest common divisor D ( p, q ) = 2. There are two cases : pq odd or pq even (with p or q odd). In the first case we choose T (2 , b ) and in the secondcase the minimal choice is T (1 , b ) or T (2 , b ) but T (2 , b ) is also suitable,the important point being that D (2 p, q ) = 2. On the covering T (2 , b ), thepull-back of our closed line (cid:96) p,q in T (1 , b ) is the union of two distinct closedlines in T (2 , b ). Coming back to the k -partition, the lifting to T (2 , b ) leadsto a (2 k )-partition. This ends the proof of the proposition. Remark 5.2
When p and q are not mutually prime, our constructions lead, as explainedin [2] to D ( p, q ) connected closed lines, where D ( p, q ) is the greatest commondivisor of p and q . The second subfigure of Figure 4 corresponds to the case p = 4 and q = 2 .To understand the point, take the closure of R ( p, q ) (see (1.1) ) and considerthe intersection of the lines of equation y = − x + c ( c ∈ Z ) with R ( p, q ) . Ifwe project on the corresponding torus and look at the number of connectedcomponents obtained on the torus, then we observe that this number is D ( p, q ) (see the second subfigure of Figure 4 which has two components). When D ( p, q ) = 1 , we get a single closed line of the torus. After a suitable dilation,we can then come back to T (1 , b ) .When D ( p, q ) (cid:54) = 1 , it is not possible to find a continuous closed line on thetorus without self-intersection with winding pair ( p, q ) . We deduce from Propositions 3.2, 4.1 and 5.1 that, if b < k ( k odd), then anyminimal k -partition can be lifted into a (2 k )-partition of T (2 , b ) with thesame energy L k ( T (1 , b )). We need to look at the spectrum of the Laplacian8n the 4-covering T (2 , b ) and to determine under which condition the (2 k )-th eigenvalue is Courant sharp. The eigenvalues are given by π ( (cid:96) + m /b ).If b < k , the (2 k ) − th eigenvalue corresponds to m = 0 and (cid:96) = k , and weare in a Courant sharp situation. Theorem 2.1 implies that π k = L k ( T (2 , b )) ≤ L k ( T (1 , b )) . Having in mind (3.1), this ends the proof of the theorem in the odd case.
Remark 6.1
The ideas in the proof might lead to results concerning minimal partitions forother ”thin” compact surfaces. b is irrational. We recall (see after Theorem 2.2) that on T (1 , b ), the associated eigenvaluesare given by λ m,n (1 , b ) = 4 π ( m + n b ) . (7.1)If m, n > b is irrational, then we have multiplicity 4. Followingsome ideas which we presented already in [5] for rectangles and the disk wehave the following result. Theorem 7.1
Suppose b is irrational. If min( m, n ) ≥ , then there is no Courant sharppair ( u, λ m,n ) . The proof is based on the following
Proposition 7.2
For m, n > any eigenfunction u corresponding to λ m,n has at most mn nodal domains. Moreover the only eigenfunctions with exactly mn nodaldomains have the form cos(2 πmx + θ ) cos(2 πn yb + θ ) for some constants θ and θ . The other eigenfunctions have D ( m, n ) nodal domains, where D ( m, n ) is the greatest common divisor of m and n . Proof of the proposition
We first observe that a general eigenfunction associated with λ m,n can be9ritten in the form: u = µ (cid:16) cos 2 πmx cos(2 πn yb + θ ) + λ sin 2 πmx cos(2 πn yb + θ ) (cid:17) , (7.2)with µ (cid:54) = 0.Note that it is only here that we use the fact that b is irrational. By rotation,we can reduce to the case when θ = 0 and we write θ = θ .Then after dilation and rotation, the proof is based on the following lemma: Lemma 7.3
Except when λ = 0 or θ ≡ π mod ( π ) , the nodal set of the function u λ,θ :=cos 2 πx cos(2 πy + θ ) + λ sin 2 πx sin 2 πy has no critical zero. Let us look at the critical zeroes of this functions. They should satisfy:cos 2 πx cos(2 πy + θ ) + λ sin 2 πx sin 2 πy = 0 , − sin 2 πx cos(2 πy + θ ) + λ cos 2 πx sin 2 πy = 0 , − cos 2 πx sin(2 πy + θ ) + λ sin 2 πx cos 2 πy = 0 . (7.3)We assume λ (cid:54) = 0. Suppose that this system has a solution. The two firstequations imply cos(2 πy + θ ) = 0 and sin 2 πy = 0. This implies cos θ = 0.Hence, when cos θ (cid:54) = 0, our function u λ,θ has no critical zero. Lemma 7.4
For λ (cid:54) = 0 , θ = 0 , and cos θ (cid:54) = 0 , the nodal partition of the function u of (7.2) has D ( m, n ) components. In each connected component of the set A := { ( λ, θ ) | λ (cid:54) = 0 , cos θ (cid:54) = 0 } in R the number of nodal domains is constant. Hence it is enough to determinethis number for one specific pair ( λ, θ ) in each component of A . It is enoughto consider λ = ± θ ≡ π ), where the computation of the numberof nodal domains is immediate (see Remark 5.2) and equal to 2 D ( m, n ).Note that when cos θ = 0, we get a product u λ,θ := sin 2 πny ( λ sin 2 πmx ± cos 2 πmx )which has 4 mn nodal domains. Remark 7.5
This is not clear for the case that b is rational, since then higher multiplici-ties could occur and we do not know how to exclude the possibility of a highernumber of nodal domains in higher dimensional eigenspaces. roof of Theorem 7.1 We give two alternative proofs (the second is geometric and inspired byarguments developed in [5]):
Proof 1
If inf( n, m ) ≥
1, then λ m,n = λ k ( n,m ) with k ( m, n ) ≥ mn + 2 m +2 n −
2. This is obtained by just adding the multiplicities of the eigenval-ues λ m (cid:48) ,n (cid:48) with m (cid:48) ≤ m , n (cid:48) ≤ n , ( m (cid:48) , n (cid:48) ) (cid:54) = ( m, n ). On the other hand,Proposition 7.2 says that any eigenfunction has at most 4 mn domains (ifinf( m, n ) ≥ Proof 2
According to Proposition 7.2 it is enough to consider eigenfunc-tions in the form sin(2 πmx + θ ) sin(2 πn yb + θ ) and to show that it cannotcorrespond to a Courant sharp case. Consider for simplicity the situationthat m = 1 = n and θ = θ = 0. Then (up to a rotation) the eigenfunctionis given by u , = sin 2 πx sin(2 πy/b ). The zeros are given by the zeros of thesines. In particular we can for instance consider the zero given by y = b/ y = 0. Consider the P = { ( x, y ) ∈ T (1 , b ) | < y < b/ } and P = { ( x, y ) ∈ T (1 , b ) b/ < y < b } and consider N i = { ( x, y ) ∈ P i | ( x, y ) ∈ N ( u , ( x, y )) } ,where N ( u ) denotes the zeroset of u . Suppose we have a minimal partitioncorresponding to this eigenfunction. Then we can rotate for instance N , sothat the zeros x = 0 , x = 1 / N fixed. The associatedpartition will still have the same energy. But this cannot correspond to aminimal partition since the equal angle property does not hold; see also [5].This argument extends to arbitrary m, n > θ , θ ). (cid:50) Remark 7.6
There exists < b < sufficiently close to , so that, for each irrational b satisfying: b < b < , only the first and the second eigenvalue togetherwith their eigenfunctions are Courant sharp pairs. This follows by counting.Remember b < . The eigenvalues all have multiplicity or . Suppose n = 0 then u m, has m nodal domains. So Courant sharpness can occur only for λ m, = λ m . This will not be the case if | − b | is small since then λ ,n willbe eventually be below λ m, hence λ m, > λ m . The case m, n ≥ has beentreated above. Acknowledgements
Thanks to P. de Soyres for his help for the pictures. The second authorhad helpful discussions with Frank Morgan during the Dido conference inCarthage 2010. 11 eferences [1] V. Bonnaillie-No¨el, B. Helffer and T. Hoffmann-Ostenhof. Spectral min-imal partitions, Aharonov-Bohm hamiltonians and application.
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B. Helffer: D´epartement de Math´ematiques, Bat. 425, Univer-sit´e Paris-Sud, 91 405 Orsay Cedex, France.email: [email protected]. Hoffmann-Ostenhof: Department of Theoretical Chem-istry, 1090 Wien, W¨ahringerstrasse 17, Austriaemail: [email protected]
A Pictures12igure 2: A 3-partition of the torus without critical point.Figure 3: The lifted 3-partition on the four-fold covering of the torus.13igure 4: (p=3, q=2) and (p=4, q=2)