## Abstract

Given a bounded open set in (or in a Riemannian manifold), and a partition of *Ω* by *k* open sets *ω*_{j}, we consider the quantity , where *λ*(*ω*_{j}) is the ground state energy of the Dirichlet realization of the Laplacian in *ω*_{j}. We denote by ℒ_{k}(*Ω*) the infimum of over all *k*-partitions. A minimal *k*-partition is a partition that realizes the infimum. Although the analysis of minimal *k*-partitions is rather standard when *k*=2 (we find the nodal domains of a second eigenfunction), the analysis for higher values of *k* becomes non-trivial and quite interesting. Minimal partitions are in particular spectral equipartitions, i.e. the ground state energies *λ*(*ω*_{j}) are all equal. The purpose of this paper is to revisit various properties of nodal sets, and to explore if they are also true for minimal partitions, or more generally for spectral equipartitions. We prove a lower bound for the length of the boundary set of a partition in the two-dimensional situation. We consider estimates involving the cardinality of the partition.

## 1. Introduction

Given a bounded open set *Ω* in (or in a Riemannian manifold), and a partition of *Ω* by *k* open sets *ω*_{j}, we consider the quantity , where *λ*(*ω*_{j}) is the ground state energy of the Dirichlet realization of the Laplacian in *ω*_{j}.

We denote by the infimum of over all *k*-partitions. A minimal *k*-partition is a partition that realizes the infimum. Although the analysis of minimal *k*-partitions is rather standard when *k*=2 (we find the nodal domains of a second eigenfunction), the analysis for higher values of *k* becomes non-trivial and quite interesting. Minimal partitions are in particular spectral equipartitions, i.e. the ground state energies *λ*(*ω*_{j}) are all equal.

The purpose of this paper is to revisit various properties of nodal sets, and to explore if they are also true for minimal partitions, or more generally for spectral equipartitions. In §3, we prove a lower bound for the length of the boundary set of a partition in the two-dimensional situation. In §4, we consider estimates involving the cardinality of the partition.

## 2. Definitions and notations

### (a) Spectral theory

Let *Ω* be a bounded domain in , or a compact Riemannian surface, possibly with boundary ∂*Ω*, which we assume to be piecewise *C*^{1}. Let *H*(*Ω*) be the realization of the Laplacian, or of the Laplace–Beltrami operator, −*Δ* in *Ω*, with Dirichlet boundary condition (*u*|_{∂Ω}=0). Let {*λ*_{j}(*Ω*)}_{j≥1} be the increasing sequence of the eigenvalues of *H*(*Ω*), counted with multiplicity. The eigenspace associated with *λ*_{k} is denoted by *E*(*λ*_{k}).

A ground state *u*∈*E*(*λ*_{1}) does not vanish in *Ω* and can be chosen to be positive. On the contrary, any non-zero eigenfunction *u*∈*E*(*λ*_{k}),*k*≥2, changes sign in *Ω*, and hence has a non-empty zero set or *nodal set*,
2.1The connected components of *Ω*\*N*(*u*) are called the *nodal domains* of *u*. The number of nodal domains of *u* is denoted by *μ*(*u*).

Courant's nodal domain theorem is the following.

### Theorem 2.1 (Courant)

*Let k≥1, and let E(λ*_{k}) *be the eigenspace of H(Ω) associated with the eigenvalue λ*_{k}. *Then, ∀u∈E(λ*_{k}*)\{0}, μ(u)≤k.*

Except in dimension 1, the inequality is strict in general. More precisely, we have the following.

### Theorem 2.2 (Pleijel)

*Let Ω be a bounded domain in* . *There exists a constant k*_{0} *depending on Ω, such that if k≥k*_{0}, *then*

Both theorems are proved in [1]. The main points in the proof of Pleijel's theorem are the Faber–Krahn inequality and the Weyl asymptotic law. The Faber–Krahn inequality states that, for any bounded domain *ω* in ,
2.2where *A*(*ω*) is the area of *ω*, and **j** is the least positive zero of the Bessel function of order 0 (**j**∼2.4048). Weyl's asymptotic law for the eigenvalues of *H*(*ω*) states that
2.3

Let be the maximum value of *μ*(*u*) when *u*∈*E*(*λ*_{k})\{0}. Combining the results of Faber–Krahn and Weyl, we obtain
2.4

### Remark 2.3

Pleijel's theorem extends to bounded domains in , and more generally to compact *n*-manifolds with boundary, with a constant *γ*(*n*)<1 replacing 4/**j**^{2} in the right-hand side of (2.4) [2,3]. It is also interesting to note that this constant is independent of the geometry.

### Remark 2.4

It follows from Pleijel's theorem that the equality can only occur for finitely many values of *k*. The analysis of the equality case is very interesting. We refer to [4] for more details.

### Remark 2.5

In dimension 1, counting the nodal domains of an eigenfunction of a Dirichlet Sturm–Liouville problem in some interval [*a*,*b*] is the same as counting the number of zeroes of the eigenfunction. An analogue in dimension 2 is to consider the length of the nodal set of eigenfunctions instead of the number of their nodal domains. We return to this question in §3.

### (b) Partitions

For this section, we refer to [4]. Let *k* be a positive integer. A (weak) *k*-*partition* of the open bounded set *Ω* is^{1} a family of pairwise disjoint sets such that . We denote by the set of *k*-partitions such that the domains *D*_{j} are open and connected.

Given , we define the *energy* of the partition as
2.5where *λ*(*D*_{j}) is the ground state energy of *H*(*D*_{j}). We now define the number as
2.6

A partition is called *minimal* if

### Example

The nodal domains of an eigenfunction *u*∈*E*(*λ*)\{0} of *H*(*Ω*) form a *μ*(*u*)-partition of *Ω* denoted by . Such a partition is called a *nodal partition*.

It turns out that , and that minimal two-partitions are nodal partitions. The situation when *k*≥3 is more complicated, and more interesting [4].

A partition is called *strong* if
2.7

The *boundary set* of a strong partition is the closed set
2.8

The set of *regular k-partitions* is the subset of strong

*k*-partitions in whose boundary set satisfies the following properties.

(i) The set

*N*is locally a regular curve in*Ω*, except possibly at finitely many points {*y*_{i}}∈*N*∩*Ω*, in the neighbourhood of which*N*is the union of*ν*(*y*_{i}) smooth semi-arcs at*y*_{i},*ν*(*y*_{i})≥3.(ii) The set

*N*∩∂*Ω*consists of finitely many points {*z*_{j}}. Near the point*z*_{j}, the set*N*is the union of*ρ*(*z*_{j})≥1 semi-arcs hitting ∂*Ω*at*z*_{j}.(iii) The set

*N*has the equal angle property. More precisely, at any*interior singular point**y*_{i}, the semi-arcs meet with equal angles; at any*boundary singular point**z*_{j}, the semi-arcs form equal angles together with the boundary ∂*Ω*.

### Example

A nodal partition provides an example of a regular partition, and the boundary set coincides with the nodal set *N*(*u*). Note that for a regular partition, the number *ν*(*y*_{i}) of semi-arcs at an interior singular point may be odd, whereas it is always even for a nodal partition.

Let us now introduce the following definition.

### Definition 2.6

We call *spectral equipartition* a strong *k*-partition such that for *i*=1,…,*k*. The number is called the *energy* of the equipartition.

### Example

Nodal partitions provide examples of spectral equipartitions.

### (c) Euler formula

Let *Ω*⊂*M* be a bounded domain with piecewise smooth boundary ∂*Ω*. Let be a regular closed set (in the sense of §2*b*, properties (i)–(iii)) such that the family of connected components of *Ω*\*N* is a regular, strong partition of *Ω*. Recall that for a singular point *y*∈*N*∩*Ω*, *ν*(*y*) is the number of semi-arcs at *y*, and that for a singular point *z*∈*N*∩∂*Ω*, *ρ*(*z*) is the number of semi-arcs at *z*, not counting the two arcs contained in ∂*Ω*. Let denote the set of singular points of , both interior or boundary points, if any. We define the *index* of a point to be
2.9We introduce the number to be
2.10For a regular strong *k*-partition of *Ω*, we have Euler's formula,
2.11

We refer to [8] for a combinatorial proof of this formula in the case of an open set of . One can give a Riemannian proof using the global Gauss–Bonnet theorem. For a domain *D* with piecewise smooth boundary ∂*D* consisting of piecewise *C*^{1} simple closed curves , with corners {*p*_{i,j}} (*i*=1,…,*n* and *j*=1,…,*m*_{i}) and corresponding interior angles *θ*_{i,j}, we have
2.12where
In this formula, *k* is the geodesic curvature vector of the regular part of the curve *C*_{i}, and *ν*_{D} is the unit normal to *C*_{i} pointing inside *D*.

To prove (2.11), it suffices to sum up the Gauss–Bonnet formulae relative to each domain *D*_{j} and to take into account the following facts:

— the integrals of the Gaussian curvature over the

*D*_{j}'s add up to the integral of the Gaussian curvature over*Ω*,— cancellations occur when adding the integrals of the geodesic curvature over the curves bounding two adjacent

*D*_{j}(the unit normal vectors point in opposite directions), whereas they add up to give the integral of the geodesic curvature over the boundary of*Ω*, and— there are contributions coming from the angles associated with the singular points of

*N*and, when summed up, these contributions yield the second term in the left-hand side of (2.11).

Note that the proof of (2.11) does not use the fact that the semi-arcs meet at the singular points of *N* with equal angles.

## 3. Lower bounds for the length of the boundary set of a regular equipartition

### (a) Introduction

Let be a regular spectral equipartition with energy . The boundary set of the partition consists of singular points inside *Ω*, of singular points on ∂*Ω*, of *C*^{1} arcs which bound two adjacent domains of the partition, and of arcs contained in ∂*Ω*. We define the length of the boundary set by the formula
3.1where ℓ denotes the length of the curves. Note that and that .

Here, we investigate lower bounds for in terms of the energy and the area *A*(*Ω*). As a matter of fact, we show that the methods introduced in [9–11] apply to regular spectral equipartitions, and hence to minimal partitions. We provide three estimates.

1. The first estimate holds for plane domains, and follows the method of Brüning & Gromes [9].

2. The second estimate applies to a compact Riemannian surface (with or without boundary), and follows the method of Savo [11].

3. The third estimate is a local estimate based on the method of Brüning [10].

Let be a regular spectral equipartition with energy . Let *R*(*D*_{i}) be the inner radius of the set *D*_{i}. Recall that **j** denotes the least positive zero of the Bessel function of order 0.

### (b) The method of Brüning and Gromes

Here *Ω* is a bounded domain in , with piecewise *C*^{1} boundary. We sketch only a method that relies on the following three inequalities.

1. The monotonicity of eigenvalues and the characterization of the ground state imply that 3.2

2. The Faber–Krahn inequality and the isoperimetric inequality imply that 3.3

3. The generalized Féjes–Toth isoperimetric inequality [9, Hilfssatz 2] asserts that, for 1≤

*i*≤*k*, 3.4

Using that *χ*(*D*_{i})≤1, we immediately see that the function *r*→*r*ℓ(∂*D*_{i})−*χ*(*D*_{i})*πr*^{2} is non-decreasing for 2*πr*≤ℓ(∂*D*_{i}). Using inequalities (3.2) and (3.3), it follows that one can substitute to *R*(*D*_{i}) in (3.4) and obtain
3.5Summing up the inequalities (3.5), for 1≤*i*≤*k*, we obtain
Using Euler's formula (2.11), we conclude that
We have proved the following.

### Proposition 3.1

*Let Ω be a bounded open set in* *and let* *be a regular spectral equipartition of Ω. The length* *of the boundary set of* *is bounded from below in terms of the energy* . *More precisely*,
3.6

Note that (3.6) is actually slightly better than the estimate in [9] which does not take into account the term when is the nodal partition for an eigenfunction *u* associated with the eigenvalue *Λ*. This fact is suggested in Savo [11].

### (c) The method of Savo

Here, we follow the method of Savo [11], and keep the same notations and assumptions. We sketch the proof in the case with boundary as it is not detailed in Savo [11]. Here, *Ω* is a compact Riemannian surface with boundary. We denote the Laplace–Beltrami operator by *Δ* and the Gaussian curvature by *K*. We write *K*=*K*_{+}−*K*_{−} (the negative and positive parts of the curvature).

We assume that numbers *α*≥0 and *D* are given such that
where *δ*(*Ω*) is the diameter of *Ω*. Finally, we define the numbers
and
We recall the following results from Savo [11].

### Lemma 3.2

*Let Ω be a compact Riemannian surface with piecewise C*^{1} *boundary. Then*
3.7*where λ*(*Ω*) *is the ground state energy of the Dirichlet realization of the Laplacian in Ω*, *and R*(*Ω*) *the inner radius of Ω*.

This is proposition 3 in Savo [11, p. 137]. Note that when *M* is flat and *Ω* is simply or doubly connected, we recover Polya's inequality [12] which reads
3.8

### Lemma 3.3

*Let Ω be a compact Riemannian surface with piecewise C*^{1} *boundary. Then*

This is lemma 10 in Savo [11, p. 141] using *λ*(*Ω*) instead of *λ*.

### Lemma 3.4

*Let Ω be a compact Riemannian surface with piecewise C*^{1} *boundary. Assume that B*(*Ω*)<0. *Then*

This is lemma 11 in Savo [11, p. 141], which relies on Dong's paper [13]. Note that the second inequality follows from lemma 3.2.

Let us now proceed with the lower estimate of when *Ω* is a Riemannian surface with boundary.

### Proposition 3.5

*Let Ω be a compact Riemannian surface with piecewise C*^{1} *boundary. The length* *of the boundary set of a regular spectral equipartition* *with energy* *satisfies the inequality*
3.9

### Proof.

The proof follows the ideas in Savo [11] closely. Because Savo does not provide all the details for the case with boundary, we provide them here. Lemma 3.2 applied to each *D*_{j} gives
Because *λ*(*D*_{j})=*Λ* for all *j*, summing up in *j*, we find that
3.10Call *T* the second term in the right-hand side of the preceding inequality and define the sets

By lemma 3.3, we have
3.11Using the definition of *B*(*D*_{j}), we find that
and hence, using Euler's formula (2.11),
On the other hand, we have
and we can estimate the last term in the right-hand side using lemma 3.4. Namely
Finally, we obtain the following estimate for *T*:
Using (3.10), it follows that
3.12This proves the proposition. □

### (d) A loose local lower estimate for

For the sake of simplicity, we now assume that *Ω* is a bounded domain in , with piecewise *C*^{1} boundary. We also assume that we are given some point *x*_{0}∈*Ω*, some radius *R* and some positive number *ρ*, small with respect to *R*, such that *B*(*x*_{0},*R*+*ρ*)⊂*Ω*. Note that the ball *B*(*x*_{0},*R*) could be replaced by any regular domain.

#### (i) A local estimate à la Brüning–Gromes: eigenvalues

### Lemma 3.6

*Let λ be an eigenvalue of H*(*Ω*), *and let u*∈*E*(*λ*) *be a non-zero eigenfunction associated with λ. If λr*^{2}>**j**^{2}, *then any disc B*(*x*,*r*)⊂*Ω contains at least a point of the nodal set N*(*u*).

This follows immediately from the monotonicity of the Dirichlet eigenvalues with respect to domain inclusion.

### Lemma 3.7

*Let λ be an eigenvalue of H*(*Ω*), *assumed to be large enough. Let r*>0 *be such that* 0<*r*≤*ρ*<*R*/10, *and λr*^{2}>4**j**^{2}. *Then, there exists a family of points* {*x*_{1},…,*x*_{N}} *such that*

(1)

*For*1≤*j*≤*N*,*x*_{j}∈*N*(*u*)∩*B*(*x*_{0},*R*−*r*/2).(2)

*The balls B*(*x*_{j},*r*/2), 1≤*j*≤*N*,*are pairwise disjoint and contained in B*(*x*_{0},*R*)⊂*Ω*.(3)

*We have the inclusion*.(4)

*The number N satisfies r*^{2}*N*≥0.2*R*^{2}.

### Proof.

(*a*) Consider the ball *B*(*x*_{0},*R*−*r*) and take *y*_{1},*y*_{2} to be the endpoints of a diameter of the closed ball. Because *r*≤*ρ*<*R*/10 and *r*^{2}*λ*>4**j**^{2}, we have that *B*(*y*_{i},*r*/2)⊂*B*(*x*_{0},*R*−*r*/2)⊂*Ω* and *B*(*y*_{i},*r*/2)∩*N*(*u*)≠∅. Choose *x*_{i} in *B*(*y*_{i},*r*/2)∩ *N*(*u*). Then, *x*_{i}∈*N*(*u*)∩*B*(*x*_{0},*R*−*r*/2), *B*(*x*_{1},*r*/2)∩*B*(*x*_{2},*r*/2)=∅ and *B*(*x*_{i},*r*/2)⊂ *B*(*x*_{0},*R*)⊂*Ω*.

(*b*) Take a maximal element {*x*_{1},…,*x*_{N}} (with respect to inclusion) in the set
so that the family {*x*_{1},…,*x*_{N}} satisfies (1) and (2).

*We claim that (3) holds*. Indeed, otherwise we could find *y*∈*B*(*x*_{0},*R*−*r*) with *d*(*x*_{i},*y*)≥2*r*, for 1≤*i*≤*N*. Because *B*(*y*,*r*/2)∩*N*(*u*)≠∅, we would find some *z*∈*B*(*x*_{0},*R*−*r*/2)∩*N*(*u*)∩*B*(*y*,*r*/2) such that . This would contradict the maximality of the family.

(*c*) Assertion (3) implies that and since *r*≤*ρ*<*R*/10, we get *r*^{2}*N*≥(0.9)^{2}(1/4)*R*^{2}. The lemma is proved. □

Recall that *N*(*u*) consists of finitely many points and finitely many *C*^{1} arcs with finite length.

### Lemma 3.8

*Let {x*_{1},…,*x*_{N}} *be a maximal family as given by lemma 3.7. Assume that r*^{2}*λ*<16**j**^{2}. *Then, there exists no nodal curve γ*⊂*N*(*u*) *which is simply closed and contained in any of the balls B*(*x*_{j},*r*/4), 1≤*j*≤*N*.

### Proof.

Indeed, otherwise, there would be a nodal domain contained in one of the balls *B*(*x*_{j},*r*/4) and hence we would have *r*^{2} *λ*≥16**j**^{2}. □

We can now prove the following local estimate.

### Proposition 3.9

*Let λ be an eigenvalue of H*(*Ω*), *assumed to be large enough. Let u be a non-zero eigenfunction associated with λ. Then, the length of the nodal set N*(*u*) *inside B*(*x*_{0},*R*) *is bounded from below by* .

### Proof.

Choose (*r*,*λ*) so that 4**j**^{2}<*r*^{2}*λ*<16**j**^{2}, with *r*≤*ρ*<*R*/10. By lemma 3.7, the *N* balls *B*(*x*_{j},*r*/4) are pairwise disjoint with centre on *N*(*u*). By lemma 3.8, the length of *N*(*u*)∩*B*(*x*_{j},*r*/4) is at least *r*/2. It follows that
3.13and the result follows in view of the estimates *r*^{2}*N*≥0.2*r*^{2} and *r*^{2}*λ*<16**j**^{2}. □

### Remark 3.10

Proposition 3.9 can be generalized to the case of a compact Riemannian surface with or without boundary. In that case, one needs to consider balls with radii less than the injectivity radius of the surface, and replace the Faber–Krahn inequality by a local Faber–Krahn inequality, using the fact that the metric can be at small scale compared with a Euclidean metric (see [10] for more details).

#### (ii) A local estimate à la Brüning–Gromes: spectral equipartitions

The above proof applies to a regular spectral equipartition of energy *Λ*. It is enough in the statements to replace the nodal set *N*(*u*) of *u* by the boundary set of the partition . We just rewrite the first statement.

### Lemma 3.11

*Let Λ be the energy of a regular spectral equipartition. If Λr*^{2}>**j**^{2}, *then any disc B*(*x*,*r*)⊂*Ω contains at least one point of boundary set of the partition*.

This follows immediately from the monotonicity of the Dirichlet eigenvalues with respect to domain inclusion.

## 4. Estimates involving the cardinality of the partitions

Let be a partition of *Ω*. We call the number *k* the *cardinality* of the partition, and we denote it by .

### (d) Estimates on the energy and on

### Proposition 4.1

(i)

*Let Ω be a bounded open subset of*.*The energy**of a partition**of Ω satisfies the inequality*4.1*In particular, for any k*≥1,*we have the inequality*4.2(ii)

*Let Ω be a bounded open subset on a compact Riemannian surface. Then*4.3

### Proof.

Assertion (i) is an immediate consequence of the Faber–Krahn inequality (2.2). To prove assertion (ii), we use the fact that on a general compact surface *M*, we have the following asymptotic isoperimetric and Faber–Krahn inequalities (which actually hold in arbitrary dimension).

### Lemma 4.2 ([3, lemma II.15, p. 528])

*Let* (*M*,*g*) *be a compact Riemannian surface. For any ϵ*>0, *there exists a positive number a*(*M*,*g*,*ϵ*) *such that for any regular domain ω*⊂*M with area A*(*ω*) *less than or equal to a*(*M*,*g*,*ϵ*),
*and*
*where ω** *is a Euclidean disc of area A*(*ω*).

Let be a partition of *Ω*. Let
The number of elements of this set is bounded by
4.4For any *i*∉*J*_{ϵ}, we can write
and hence, provided that is large enough,
As a consequence, we obtain that
We can now let *ϵ* tend to zero to get the estimate (4.3). □

### Remarks

(1) We point out that the lower bounds in the proposition only depend on the area of

*Ω*, not on its geometry.(2) Similar inequalities on can also be deduced from [2], when

*Ω*is a bounded domain in a simply connected surface*M*with Gaussian curvature*K*, such that*Ω*⊂*Ω*_{0}, a simply connected domain satisfying . Let us mention two particular cases.(a) If

*M*is a simply connected surface with non-positive curvature, then according to Peetre [2],*λ*(*Ω*)*A*(*Ω*)≥*π***j**^{2}for any bounded domain*Ω*and we conclude that for all*k*≥1, as in the Euclidean case.(b) If

*M*is the standard sphere, then according to Peetre [2], for any domain*D*, and one can conclude that, for any domain*Ω*, for all*k*≥1.

### Remark 4.3

Given a *k*-partition of *Ω*, one can also introduce the energy , and define the number by taking the infimum of the *Λ*_{1}-energy over all *k*-partitions. An easy convexity argument shows that the inequalities (4.1)–(4.3) hold with and replaced by and respectively. For example, we have the inequalities
and

For bounded domains in , one can also give an asymptotic upper bound for . More precisely, one has the following.

### Property 4.4

For any regular bounded open subset of ,
4.5where Hexa_{1} is the regular hexagon in , with area 1.

This can be seen by considering the hexagonal tiling in the plane, with hexagons of area *a* and the partition of *Ω* given by taking the union of the hexagons contained in *Ω*, whose number is asymptotically when *a* tends to zero.

The inequalities (4.5) and (4.3) motivate the following two conjectures^{2} for bounded domains in . They were proposed and analysed in recent years (see [4,14–17]). The first one is the following.

### Conjecture 4.5

The limit of as exists.

For the second one, this limit is given more explicitly.

### Conjecture 4.6

The second conjecture says, in particular, that the limit only depends on the area of *Ω*, not on its geometry (provided *Ω* is a regular domain).

It is explored numerically in [14] why the second conjecture looks reasonable. Note that Caffarelli & Lin [16] mention conjecture 4.6 in relation with . From this point of view, the recent numerical computations by Bourdin *et al.* [15] for the asymptotic structure of the minimal partitions for are very enlightening (figure 1). Remark 4.3 shows that (4.3) should be a strict inequality.

### (b) Asymptotics of the length of the boundary set of minimal regular *k*-equipartitions for *k* large

Here, we consider only bounded open domains *Ω* in . In this case, conjecture 4.6 leads to a natural ‘hexagonal conjecture’ for the length of the boundary set.

### Conjecture 4.7

4.6where ℓ(Hexa_{1}) is the length of the boundary of the hexagon of area 1,

For regular spectral equipartitions of the domain *Ω*, inequalities (3.6) and (4.1) yield
4.7Assuming that *χ*(*Ω*)≥0, we have the uniform lower bound
4.8

### Remark

Assume that *χ*(*Ω*)≥0, and that all the subdomains *D*_{i} in the regular equipartition satisfy *χ*(*D*_{i})≥0 as well. Then, Polya's inequality (3.8) yields the sharper inequality
4.9

The following statement is a particular case of theorem 1B established by Hales [18] in his proof of Lord Kelvin's honeycomb conjecture (see also [19]).

### Theorem 4.8

*Let Ω be a relatively compact open set in* *and let* *be a regular finite partition of Ω. Then
*4.10

In order to optimize the use of this theorem, we consider , and apply the theorem to a dilated partition. If we dilate by *t* the length is multiplied by *t* and the area by *t*^{2}. So, we take , and we obtain the following.

### Corollary 4.9

*For any regular partition* *of a bounded open subset Ω of*
4.11

### Proposition 4.10

*Let Ω be a regular bounded domain in* .

(i)

*For k*≥1,*let**be a minimal regular k-equipartition of Ω. Then*4.12(ii)

*If χ*(*Ω*)≥0,*then for any regular spectral equipartition, we have the universal estimate*4.13

### Remarks

(a) Recall that

*λ*(Hexa_{1})∼18,5901 and that*λ*(Disc_{1})=*π***j**^{2}∼18,1680. It follows that (*π***j**^{2}/*λ*(Hexa_{1}))^{1/2}∼0,989, so that the right-hand side of (4.12) is very close to the right-hand side of (4.6), the hexagonal conjecture for the length.(b) Asymptotically, when we consider minimal regular

*k*-equipartition , inequality (4.13) is weaker than (4.12) but it is universal, and independent of the asymptotics of the energy of the partition.

### Proof of the proposition.

(i) Let be a regular equipartition of *Ω*. The Faber–Krahn inequality (2.2) gives
hence, combining with (4.11),
Let be a minimal regular *k*-equipartition of *Ω*. Applying (4.2), we obtain
Using the upper bound for given by (4.5), we obtain the following asymptotic inequality for the length of a minimal regular *k*-equipartition:

(ii) Assume that *χ*(*Ω*)≥0. Indeed, by (3.6), we have
and hence,
The same proof as for assertion (i) gives, for any regular spectral equipartition,
Inequality (4.13) follows. □

### Remark

Assume now that is the nodal partition of some *k*th eigenfunction *u*_{k} of *H*(*Ω*). Assume furthermore that *χ*(*Ω*)≥0. The same reasoning as above gives
Hence,
This inequality is less convincing, because we have no lower bound for the right-hand side. Indeed, on the round sphere , any eigenspace of the Laplacian contains eigenfunctions with only either two or three nodal domains [20].

We may however return to the initial inequality,
Assuming Polya's conjecture [21] which says that for bounded domains in , *λ*_{k}(*Ω*)≥4*πk*/*A*(*Ω*), for *k*≥1, we find that
which should be compared with (4.8).

## Footnotes

One contribution of 13 to a Theo Murphy Meeting Issue ‘Complex patterns in wave functions: drums, graphs and disorder’.

↵1 Note that we start from a very weak notion of partition. We refer to [4] for a more precise definition of classes of

*k*-partitions, and for the notion of regular representatives.↵2 The second author was informed of these conjectures by M. Van den Berg.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.