## Abstract

In this paper, we study the set of points in the plane defined by {(*x*, *y*)=(*λ*_{1}(*Ω*), *λ*_{2}(*Ω*)), |*Ω*|=1}, where (*λ*_{1}(*Ω*), *λ*_{2}(*Ω*)) are either the first two eigenvalues of the Dirichlet–Laplacian, or the first two non-trivial eigenvalues of the Neumann–Laplacian. We consider the case of general open sets together with the case of convex open domains. We give some qualitative properties of these sets, show some pictures obtained through numerical computations and state several open problems.

## 1. Introduction

Let be a bounded open set and consider the Dirichlet eigenvalue problem,
1.1defined on the Sobolev space . We will denote the eigenvalues by 0<*λ*_{1}(*Ω*)≤*λ*_{2}(*Ω*)≤⋯ (counted with their multiplicities) and the corresponding orthonormal real eigenfunctions by *u*_{i}, *i*=1,2,…. Wolf & Keller (1994) and Bucur *et al.* (1999) studied the region
which is the range of the first two Dirichlet eigenvalues of planar sets with unit area. We also refer to Levitin & Yagudin (2003) for a similar study for the first three eigenvalues. Let us begin with some elementary facts. Obviously lies in the first quadrant and within the sector 0<*x*≤*y*, because we defined the eigenvalues to be ordered. The behaviour of eigenvalues with respect to homothety (*λ*_{k}(*tΩ*)=*λ*_{k}(*Ω*)/*t*^{2}) has two consequences. First, we can also write . Then, it is clear that the region is conical with respect to the origin in the sense
(consider a homothety of ratio of the original domain and complete with a collection of small balls to reach volume 1 without changing the first two eigenvalues). Now, we can get more precise information about thanks to some important results on the low eigenvalues of the Laplacian. This region can be reduced using the famous Faber–Krahn inequality proved in Faber (1923) and Krahn (1924), which states that the ball minimizes *λ*_{1} among all planar domains with the same area. We can write this result as
where *j*_{n,k} denotes the *k*th positive zero of the Bessel function *J*_{n} and denotes the ball of unit area. Equality holds if and only if *Ω* is a ball (up to a set of zero capacity). For the second eigenvalue, we know that the minimum is attained by two balls of equal area. This result is due to Krahn and has been rediscovered by Szegö, cited in Pólya (1955) and some other authors, see Henrot (2006) for more details. It can be written as
The quotient *λ*_{2}/*λ*_{1} is maximized at the ball (cf. Ashbaugh & Benguria 1991), or equivalently,
It is also known that the set is convex in the *x*-direction,
and in the *y*-direction
and is closed (cf. Bucur *et al.* 1999). Moreover, the tangents at the extremal points are vertical (at the ball) and horizontal (at two balls) (cf. Wolf & Keller 1994). The above results show that the only unknown part of the set is the lower part, the curve joining the points corresponding to one ball and two balls. In figure 1, we have determined numerically this curve with the same procedure as in Wolf & Keller (1994), solving a minimization problem with a convex combination of *λ*_{1} and *λ*_{2}. Our results were obtained using the gradient method to solve the minimization problems, as in Alves & Antunes (in preparation). The method of fundamental solutions (MFS), or in some cases an enriched version of the MFS, was used to solve the problems. In both cases, for a fixed value *λ* we consider a linear combination of some functions that satisfy the partial differential equation (PDE) of the eigenvalue problem (1.1). In the MFS, these base functions are built translating the fundamental solution to some source points placed outside *Ω* and this method can produce highly accurate approximations when the domain is smooth (e.g. Alves & Antunes 2005; Barnett & Betcke 2008). However, for polygonal shapes it is convenient to enrich the space of functions with some particular solutions of the PDE, which also satisfy the null boundary conditions along a corner (cf. Antunes & Valtchev 2010). We recall the conjecture already stated in Bucur *et al.* (1999):

### Conjecture 1.1

The set is convex.

The plan of this paper is the following. In §2, we study the subset of corresponding to convex domains. We look at the case of some particular polygons, we prove that is closed, connected by arcs, and give some information on its boundary. In §3, the case of Neumann eigenvalues for general domains is considered. We describe a part of its boundary and some other qualitative properties. Finally, in §3*b* we address a similar study for the first two non-trivial Neumann eigenvalues of convex domains. We prove, in particular, that the convex domain which maximizes the second (non-trivial) Neumann eigenvalue is not a stadium.

## 2. The Dirichlet case with convex domains

In this section, we are interested in the subset of obtained for convex sets,

We start with some numerical results obtained for particular classes of convex polygons.

### (a) The case of convex polygons

In this section we present some numerical results for polygonal convex domains. We generated randomly a sample of convex polygons and calculated the first two Dirichlet eigenvalues. Our sample has 4500 triangles, 23 000 quadrilaterals and 45 000 octagons. We will see that it is convenient to distinguish two types of isosceles triangles which play different roles in the region . For this purpose, we recall a definition introduced in Antunes & Freitas (2008), but instead of the terminology used in that paper we will follow Laugesen & Siudeja (2009*a*).

### Definition 2.1

The aperture of an isosceles triangle is the angle between its two equal sides. A triangle subequilateral if it isosceles with aperture less than *π*/3, and superequilateral if it is isosceles with aperture greater that *π*/3.

In figure 2*a*, we plot the subregion of that we obtain for triangles. The numerical results indicate that the region is delimited above (and below) by the curve corresponding to superequilateral (and subequilateral) triangles. To illustrate this fact, for some of the points in the figure, we included the plot of the corresponding isosceles triangles. The minimum of *λ*_{2} over all the superequilateral triangles, is attained for the right triangle and the minimum of *λ*_{2} among the subequilateral triangles, which is also the minimum over all triangles is attained for a triangle whose aperture is approximately equal to 0.5996796. It is well known (Pólya theorem, see Henrot 2006) that the minimum of *λ*_{1} is attained for the equilateral triangle. This result is also illustrated in the figure, where it can be seen that the minimum of *λ*_{1} is attained at the intersection of the curves corresponding to subequilateral and superequilateral triangles, which is the equilateral triangle. It can be proved that the tangent to the region at the point of the equilateral triangle is vertical. In figure 2*b* we plot similar results obtained for convex quadrilaterals. We marked the rhomboidal domains with larger yellow points and the rectangles with a continuous curve in grey. It is straightforward to verify that the curve of the rectangles is defined by
2.1and the numerical results that we gathered suggest that this is also the lower part of the boundary of the region that we obtain for convex quadrilaterals.

### Conjecture 2.2

For a convex quadrilateral *Q* with unit area we have
Equality holds if *Q* is a rectangle.

The upper part of the boundary for seems to correspond to the superequilateral triangles. For the upper boundary is defined by a curve of quadrilaterals with one symmetry (having *λ*_{2}=*λ*_{3}), which connects the square to the equilateral triangle. The fact that the rectangles and some isosceles triangles appear as extremal sets has already been detected by Antunes & Freitas (2006) when studying isoperimetric bounds for *λ*_{1}(*Ω*) and by Antunes & Freitas (2008) for the spectral gap, *λ*_{2}(*Ω*)−*λ*_{1}(*Ω*).

In figure 3, we plot the results obtained for convex octagons. The regular polygons are marked with a plus symbol (+) and the ball with a large point. Also in this case, the superequilateral triangles seem to be on the upper part of the boundary. We give below in §2*b* some conditions relating to the property for a domain to be on the boundary of .

### Conjecture 2.3

The superequilateral triangles are on .

A similar conjecture was proposed in Antunes & Freitas (2008) for the spectral gap. Now we present numerical results for the lower boundary of . The results plotted in figure 3 seem to suggest that we have a continuous family of rectangles on the lower part of the boundary, but in §3, we prove that this is not true. We determined numerically that part of the boundary and show the results in figure 4. We show the boundary of region with a continuous blue line and we marked the boundary of region with a dashed red line, already identified in figure 1. We observed that for the general case of region in the neighbourhood of the ball, the domains that are in the lower part of are convex. Then, in that region the boundaries and coincide. In figure 4 we marked , the location where the two boundaries split into two distinct curves, with a red point. In figure 5, we plot a domain *Ω* with unit area for which we have .

The lower point of domain has already sparked off some study. Troesch (1973) formulated the conjecture that the convex planar domain that minimizes the second eigenvalue could be the stadium (the convex hull of two identical tangent balls). This conjecture was refuted by Henrot & Oudet (2001). However, the analytical and numerical results presented in Henrot & Oudet (2003) and Oudet (2004) show that the optimal domain is very close to the stadium, not only from the numerical point of view, but also from a geometrical point of view. With our algorithm, we obtained a numerical optimal convex domain with unit area for which we have *λ*_{2}≈37.987, which is smaller that the corresponding value of the stadium for which we have *λ*_{2}≈38.002. Our numerical optimal domain is plotted in figure 6, together with the optimal convex domain for *μ*_{2}. In §3, we provide some analytical results, namely a study of the properties of the domains that are on the boundary of region and the tangents at some extremal points.

### (b) Some qualitative results

In this section, we prove some results which describe to some extent the region . The first property is of topological nature. Using continuity of eigenvalues of convex domains with respect to Hausdorff convergence, it is easy to prove that is closed and connected by arcs. Actually, it seems to be simply connected but it is probably much harder to prove. Now, the main point is to describe the boundary of the set . We will say that a domain *Ω* is on the boundary of if the point (|*Ω*|*λ*_{1}(*Ω*),|*Ω*|*λ*_{2}(*Ω*)) lies on the boundary of . In the theorem below, we give some necessary conditions for a domain *Ω* to be on the boundary of . For the sake of simplicity, we only consider the case of *C*^{2} strictly convex domains and the case of polygons, which are certainly the most significative ones. For intermediate cases, the idea is the same, but the statements would be more complicated.

### Theorem 2.4

*The region**is unbounded, connected by arcs and closed.**Let Ω be a C*^{2}*domain, strictly convex, which is on the boundary of**and assume that λ*_{2}*(Ω) is simple. Then functions 1,|∇u*_{1}*|*^{2}*,|∇u*_{2}*|*^{2}*are linearly dependent on the boundary of Ω.**Let Ω be a polygon with N sides which is on the boundary of**. Let us denote by γ*_{k}*,k=1,…,N its sides and a*_{k}*the length of γ*_{k}*. Let us introduce the 2N plane vectors defined by*2.2*Let us assume that λ*_{2}*(Ω) is simple. Then the 2N vectors V*_{k}*,V ′*_{k}*are colinear.*

### Proof.

It is sufficient to look at the first two eigenvalues of rectangles (which are explicitly known) to conclude that region is unbounded.

Now let *P*_{1} and *P*_{2} be two arbitrary points in region . By definition of , there exist two open planar convex sets with unit area *Ω*_{1} and *Ω*_{2} for which
Now let us consider the family of convex domains defined by (Minkowski sum)
The map *t*↦*Ω*_{t} is continuous when the set of convex domains is endowed with the Hausdorff convergence (see Henrot & Pierre (2005) or Gruber (1993) for more details on this convergence). By continuity of the volume and the Dirichlet eigenvalues for the Hausdorff convergence of convex sets (see theorem 2.3.17 in Henrot 2006 for the eigenvalues), the continuous path
is contained in and connects the points *P*_{1} and *P*_{2}. We conclude that is connected by arcs. Now let us prove that the region is closed. Let *Ω*_{n} be a sequence of convex domains of area 1 such that (*λ*_{1}(*Ω*_{n}),*λ*_{2}(*Ω*_{n}))→(*x*,*y*). We use the following inequality due to Makai (1962), which holds for any convex planar domains
to claim that the sequence *Ω*_{n} has a bounded perimeter. Thus, by translation, we can assume that *Ω*_{n} stays in some bounded fixed ball. The Blaschke selection theorem (see also more generally corollary 2.2.24 in Henrot & Pierre 2005) shows that we can extract from sequence *Ω*_{n} a subsequence which converges for the Hausdorff convergence, and the result follows using continuity of the eigenvalues for this convergence for convex domains.

Now we will use the Hadamard formula of differentiation with respect to the domain (e.g. Sokolowski & Zolesio 1992; Henrot & Pierre 2005; Henrot 2006). We recall its definition. Let us consider an open set *Ω* and a function *Φ*(*t*) such that
where is the set of bounded Lipschitz maps from into itself, *I* is the identity and *V* is a deformation field. Let us denote by *Ω*_{t}=*Φ*(*t*)(*Ω*), *λ*_{k}(*t*)=*λ*_{k}(*Ω*_{t}) and by *u*_{k} a (normalized) eigenfunction of *Ω* associated to *λ*_{k}(0) in . If we assume that *Ω* is convex and *λ*_{k}(*Ω*) is simple then
2.3We will also use the formulae for the derivative of the area. Define the function *area*(*t*)=|*Ω*_{t}|; then, if *Ω* is Lipschitz, we have
2.4Therefore, if we make the product, by equations (2.3) and (2.4) we have
2.5Thus, if we make a deformation of a given convex set *Ω* through a deformation field *V* *which preserves convexity*, we define a curve starting from (*x*_{0},*y*_{0})=(|*Ω*|*λ*_{1}(*Ω*),|*Ω*|*λ*_{2}(*Ω*)) with a tangent in the direction of the vector
2.6In particular, if we are able to choose deformation fields *V* such that the vector *X*_{V} covers all possible directions, it means that *Ω* lies in the interior of .

Let us consider the strictly convex case. Then, up to first order variations, we can assume that any deformation field *V* is admissible, in the sense that it preserves convexity, see theorem 4.2.2 in Henrot (2006) for more details. Since the map *V* ↦*X*_{V} is linear, then either its range is of dimension two (and thus *Ω* lies in the interior of ) or its range is of dimension less or equal to one which means
2.7This implies that the functions 1,|∇*u*_{1}|^{2},|∇*u*_{2}|^{2} are linearly dependent on the boundary of *Ω*.

Now, let us consider the case of a polygon with *N* sides. Obviously, we can perform only a few variations if we want to preserve convexity, see theorem 7.5 in Jerison (1996) or theorem 4.2.2 in Henrot (2006) for an analysis of the Hadamard formula for non-strictly convex domains in this context. We choose here to move a side *γ*_{k} of the polygon

— either in a parallel way (which corresponds to choosing a constant deformation field

*V*⋅*n*≡1 on side*γ*_{k}).— or to rotate it from one of its vertices (which corresponds to choosing a deformation field proportional to the arc-length

*V*⋅*n*≡*t*on the side*γ*_{k}).

Actually, combining the previous deformations exactly corresponds to moving the vertices of the polygon. Each of these deformations preserves convexity and the vectors *X*_{V} defined in equation (2.6) correspond to the 2*N* vectors *V*_{k} and *V* ′_{k} defined in (2.2). If these 2*N* vectors were not collinear, we would be able, by linear combination, to make a perturbation of (|*Ω*|*λ*_{1}(*Ω*),|*Ω*|*λ*_{2}(*Ω*)) in any direction proving that polygon *Ω* is not on the boundary of . ■

### Remark 2.5

Point (2) of the previous theorem has a local character: if the boundary of a convex set *Ω* has a part *γ* which is strictly convex and if *Ω* is on the boundary of , then the linear dependence of 1,|∇*u*_{1}|^{2},|∇*u*_{2}|^{2} will hold on *γ*.

Figure 3 seems to suggest that some rectangles are on the lower boundary of , but we are going to prove that this is not true. Unfortunately, we cannot use the last part of theorem 2.4. Indeed, if we compute the eight vectors defined in (2.2) for the rectangle (0,*L*)×(0,ℓ), we get eight vectors colinear to . The reason is the following: the perturbations used in theorem 2.4 consist in moving the vertices of the polygon. The rectangles are (certainly) on the boundary of the set restricted to quadrilaterals, see figure 2; therefore, we need more sophisticated perturbations to prove the following:

### Theorem 2.6

*Except for the square, the rectangles are not on the boundary of the region* *.*

### Proof.

Let *L*>1 and define a rectangle (with unit area) by the vertices (0,0), (*L*,0), (*L*,1/*L*) and (0,1/*L*). We have
and the corresponding normalized eigenfunctions are
Now consider a pentagon obtained from the rectangle moving the point (*L*,1/(2*L*)) in the direction of the vector (1,0) (see figure 7) and define
The deformation field is
2.8Then, by equation (2.5)

The calculations for *λ*_{2} are similar leading to
Then, for this particular perturbation of the rectangle,
On the other hand, for rectangles we have (here the prime denotes the derivative with respect to *L*)
and we shall compare with
We note that
and
Then,
The perturbation that we have described above allows the point (|*Ω*|*λ*_{1},|*Ω*|*λ*_{2}) to go below the values obtained for the rectangles. ■

To illustrate the result that we have proved, in figure 8 we plot the curve of the rectangles, the unitary vector which is tangent to the curve (in blue) and the unitary vector associated to the perturbation described in the proof of theorem 2.6 (in red).

Actually, we think that the lower part of region only contains points corresponding to regular domains and thus not polygons. It can be seen as a question related to the regularity of some shape optimization problem, namely minimizing *λ*_{2} with *λ*_{1} fixed, which is beyond the scope of this paper. It is a difficult question; see Briançon (2004) for similar results. Nevertheless, to prove this claim, we can also consider a similar method to the one used for rectangles. For example, if we could prove that the lower part of is a convex curve , perturbations which consist in splitting a side in two parts as we did for rectangles would allow to find a continuous curve of perturbations with a tangent (strictly) below .

### Conjecture 2.7

The lower part of the boundary of does not contain any polygons.

The numerical results showed in figure 2 suggest that the equilateral triangle and the square are on the upper part of the boundary of the region obtained for quadrilaterals. Thus, it is an interesting question to know which is the tangent to the boundary at those points. Using simple variational methods, it is possible to prove that the tangent at the equilateral triangle is vertical. By equation (2.1) we can infer that the tangent at the square is also vertical. For the general case of region , Wolf & Keller (1994) proved that the tangent at the ball is vertical. Moreover, if we could prove that the domains minimizing *λ*_{2} with *λ*_{1} fixed are regular, this would prove that these domains are still convex, for *λ*_{1} close to the value for the ball, as regular perturbations of the ball. Therefore, the tangent of the lower part of at the point associated to the ball would be vertical.

As already mentioned, we were not able to prove, but we believe

### Conjecture 2.8

The region is simply connected.

## 3. The Neumann case

In this section, we consider the similar case of finding the range of the first two non-trivial eigenvalues in the Neumann case. Let be a bounded open set. As we want to restrict ourselves to a discrete spectrum, we will assume that the embedding *H*^{1}(*Ω*)↪*L*^{2}(*Ω*) is compact. It is well known that a Lipschitz boundary is a sufficient condition for it to hold. We consider the Neumann eigenvalue problem,
3.1defined on *H*^{1}(*Ω*). Let us denote the eigenvalues by 0=*μ*_{0}≤*μ*_{1}(*Ω*)≤*μ*_{2}(*Ω*)≤… (counted with their multiplicities) and the corresponding orthonormal real eigenfunctions by and *u*_{i}, *i*=1,2,….

### Remark 3.1

If *Ω* is disconnected, for example if , where *N*≥2 and *Ω*_{k} are disjoint bounded connected open sets, we have
because we can choose an eigenfunction of *Ω* to be constant in one of the sets *Ω*_{k} and vanish in the others. In particular, any set *Ω* which is the union of three disjoint sets satisfies *μ*_{1}(*Ω*)=*μ*_{2}(*Ω*)=0 and then the corresponding point in (see below) is the origin.

Our study will focus on the first two (non-trivial) eigenvalues *μ*_{1} and *μ*_{2}. We start with the general case of a planar bounded set and then we consider the convex case.

### (a) General planar bounded sets

In this section, we will study the region
Some results that are already known help us to plot the corresponding picture. It has been conjectured by Kornhauser & Stakgold (1952) that the disc maximizes *μ*_{1} among plane domains of given area. This result has been proved by Szegö (1954) for Lipschitz simply connected domains and generalized by Weinberger (1956) to arbitrary (not necessarily simply connected) domains in any dimension. The result can be written as
3.2Moreover, *μ*_{1} is double at the ball and then the point in corresponding to the ball is on the line *μ*_{2}=*μ*_{1}. Recently, Girouard *et al.* (2010) proved that the maximum of *μ*_{2} among simply connected bounded domains is attained for two disjoint balls of equal area, which can be written as
3.3In Szegö (1954), it has been proved that for simply connected domains with unit area we have
3.4which can be written as
3.5These results and the trivial bounds *μ*_{2}≥*μ*_{1} and *μ*_{1}≥0 allow to plot a region (figure 9) which contains (at least if we restrict ourselves to simply connected domains). Now we will concern our study in knowing which part of the region of figure 9 is contained in . We denote by *αΩ* the image of *Ω* by a homothety with ratio *α*>0 and start proving an auxiliary result.

### Lemma 3.2

*Let* *and* *be two open connected disjoint sets with* . *Define* *and for each A*∈]0,1[ *let Ω be the domain* . *Then*,
*Moreover, the maximum of μ*_{2}(*Ω*) *among all possible values A*∈]0,1[ *is attained when* .

### Proof.

We know that |*αΩ*|=*α*^{2}|*Ω*|, then . Now , and thus both components do not degenerate into a single point and *Ω* has exactly two disjoint components. By remark 3.1, we have *μ*_{1}(*Ω*)=0 and . Thus,
and the lemma follows. ■

### Remark 3.3

By lemma 3.2 and inequality (3.2) it is easy to conclude that the maximum of *μ*_{2} among all disconnected sets *Ω* is attained for two balls with the same area, which is a much weaker result but related with inequality (3.3).

We will also need three classical results, the first one related to symmetry and multiplicity, the second one to symmetry and the third one to continuity of eigenvalues. The first lemma is due to in Ashbaugh & Benguria (1993, lemma 4.1)

### Lemma 3.4

*If* *is a simply connected domain with k-fold symmetry where k*≥3, *then μ*_{2}=*μ*_{1}.

The second lemma is classical, but we give a proof here for the sake of completeness.

### Lemma 3.5

*If* *is symmetric with respect to some hyperplane H, then for each eigenvalue we can find an eigenfunction which is either even or odd with respect to H*.

### Proof.

Let *x*′ denote the reflection of *x* with respect to *H*. Obviously, if *v*(*x*) is an eigenfunction (Dirichlet or Neumann) associated to an eigenvalue *λ*, the function *w*(*x*):=*v*(*x*′) is still an eigenfunction associated to *λ* and it has the same *L*^{2} norm. Therefore, the case of a simple eigenvalue is clear since we have then *w*=*v* (*v* even) or *w*=−*v* (*v* odd).

Let us now consider the case of a multiple eigenvalue. Let *v*_{1},*v*_{2},…,*v*_{m} be an orthogonal sequence of eigenvalues spanning the eigenspace *V* . For each *k*, the function *w*_{k}(*x*):=*v*_{k}(*x*′) is in *V* . Let *A* be the linear transformation mapping the *v*_{k} onto the *w*_{k}. Since *x*′′=*x*, map *A* is an involution: *A*^{2}=*Id*. Thus, and 1 or −1 is an eigenvalue of *A*. Now, we look for a combination of the *v*_{k}: such that *w*(*x*′)=*w*(*x*) or *w*(*x*′)=−*w*(*x*). This exactly corresponds to looking for an eigenvector of *A* associated to the eigenvalue 1 or −1 which is always possible. ■

### Remark 3.6

If *Ω* has two axes of symmetry and if we are interested in the second or third eigenvalue, a simple consequence is that there exists an eigenfunction which is even with respect to one of the axes of symmetry. Indeed, according to the Cayley–Hamilton theorem, the minimal polynomial of *A* divides its characteristic polynomial. Now, this minimal polynomial is either *X*^{2}−1, or *X*−1 or *X*+1. Thus, the only case where we could not find an even eigenfunction would be if the minimal polynomial is *X*+1 for both reflections with respect to the two hyperplanes. Now, it would imply that the eigenfunctions are odd with respect to the two hyperplanes. But this would imply that the eigenfunctions have (at least) four nodal domains.

The third lemma is mainly due to Chenais (1975), see also Henrot (2006, theorem 2.3.25) and Henrot & Pierre (2005, theorem 3.7.2).

### Lemma 3.7

Let *B* be a fixed compact set in and *Ω*_{n} a sequence of open subsets of *B*. Assume that the sets *Ω*_{n} are uniformly Lipschitz (in the sense that there exists a uniform Lipschitz constant for all the domains in the sequence). Assume moreover that *Ω*_{n} converge, for the Hausdorff distance, to *Ω*. Then, for every fixed *k*, *μ*_{k}(*Ω*_{n})→*μ*_{k}(*Ω*).

### Remark 3.8

More generally, the previous continuity property holds true if we can find a sequence of extension operators *P*_{n}:*H*^{1}(*Ω*_{n})↦*H*^{1}(*B*) with ∥*P*_{n}∥ (the operator norm) uniformly bounded.

Denote by *O* the origin and define the points and (figure 9). In the region , these points correspond to the ball and two identical balls, respectively. Now we define the paths
and
and prove that these paths are contained in the region . We gather in the next theorem, the main theoretical results we are able to prove concerning .

### Theorem 3.9

*We have* *and* *. More precisely, region* *contains the following sub-domains (figure *10*)
**and
**At last, the tangent of the boundary of* *at the point P*_{1} *is parallel to the second bisectrix y+x=0.*

### Proof.

We have (remark 3.1). Now let *Ω*_{a}, *a*<1/2, be the polygonal domain plotted in figure 11, which is the union of two rectangles with sides of length equal to 2 and 2*a*. For these domains, according to lemma 3.4 we have that *μ*_{2}=*μ*_{1}, because of the fourfold symmetry. Now we prove that
We know that
and then
where *v* is the test function
that satisfies . Now we have
and
Then, using , we get
Now, we have |*Ω*_{a}|=8*a*−4*a*^{2} and, therefore,
Let us fix . By continuity of the Neumann eigenvalues (apply lemma 3.7) we have
Now for *t*∈[0,1[, let *Ω*^{t} be the domain defined by which has the same type of symmetries of *Ω*_{a}, and then we have *μ*_{2}(*Ω*^{t})=*μ*_{1}(*Ω*^{t}) and again by continuity, we have
Then we have proved that . Now we prove that using lemma 3.2 twice. Let and *Ω* be the domain defined in lemma 3.2. We have and then
3.6Now take and , where *R*_{L} is a rectangle with unit area and whose length sides are *L*≥1 and 1/*L*. We know that
If *Ω* is the domain defined in lemma 3.2 we have
and then
3.7Now, by equations (3.6) and (3.7) and taking , we conclude that
and we get the conclusion.

Now we prove that we have *x*-convexity in some parts of region . Let be a rectangle with unit area for which the length of the largest side is equal to *L*. We will assume that the vertices are the points (0,0), (*L*,0), (*L*,1/*L*) and (0,1/*L*). If we have , then and . The associated eigenfunctions are
3.8In the case we have and *u*_{2} given by equation (3.8) is one of the possible eigenfunctions associated to *μ*_{2}. Now we note that
Let us introduce the domain plotted in figure 12 corresponding to the previous rectangle where we remove two segments of length *α* at *x*=*L*/2. We can check that the embedding is still compact. Indeed it is possible to make two reflections along the axis of symmetry for any function in and we easily get compactness in *L*^{2} of any bounded sequence. Thus the spectrum of the Neumann–Laplacian on is discrete. Moreover, for 0<*α*<*α*′<1/2*L*, we have
therefore, by the min–max formulae, for any eigenvalue
3.9Now is the union of two disjoint rectangles and (the corresponding eigenfunction being *u*_{2} defined in equation (3.8)). Therefore, for *k*=2 in equation (3.9) we have for any *α*, . Moreover, decreases continuously from to 0 (use remark 3.8 for the continuity). Then we conclude that
The same analysis is valid in the case and allows to prove that
Thus, we conclude that the region plotted in figure 10*a* is contained in . The curve corresponding to the rectangles is also plotted in the figure.

Now we note that the same argument can be applied for the family of domains with fourfold symmetry and two axes of symmetry (which corresponds to points on the first diagonal *μ*_{1}=*μ*_{2}), such as the domain plotted in figure 11. Let *Ω* be such a domain which satisfies *μ*_{2}(*Ω*)=*μ*_{1}(*Ω*). According to lemma 3.5 and remark 3.6, we know that there exists one eigenfunction associated to *μ*_{1}(*Ω*_{t}) which satisfies a null Neumann condition on the axis of symmetry. Then, we can add two segments on that line and we conclude that, when we increase the length of the segments, one of the eigenvalues goes to zero, and the other remains constant. The only point we need to check here is that the second eigenvalue of the disconnected domain is still *μ*_{2}(*Ω*). Actually, if there would exist an eigenfunction with a smaller eigenvalue on half of the domain, we would be able, by reflection, to construct an eigenfunction on *Ω* with the same eigenvalue, which is impossible. Then we proved that
Joining this regions and the result of theorem 3.9 we have proved that the region plotted in figure 10*b* is contained in .

Now we study the tangent of at the point *P*_{1} which corresponds to the ball. By equation (3.5) and writing we have
3.10Now to prove that the line is the tangent line at *P*_{1}, it remains to exhibit a deformation of the ball which has this behaviour. For that purpose, we still use the Hadamard formula of derivation with respect to the domain. Now we have to carry out the work for Neumann eigenvalues in the case of a *double* eigenvalue. We keep notations introduced in the proof of theorem 2.4. If a Neumann eigenvalue is simple, its derivative is given by (Henrot & Pierre 2005; Henrot 2006):
3.11Now when we deal with a double eigenvalue, it can be proved (Rousselet 1983; Henrot 2006) that *μ*_{1} and *μ*_{2} are not differentiable, but nevertheless *V* ↦(*μ*_{1}(*Ω*_{t}),*μ*_{2}(*Ω*_{t})) has directional derivatives which are precisely the two eigenvalues of the 2×2 matrix defined by (here *u*_{1},*u*_{2} are two normalized orthogonal eigenfunctions):
3.12In the case of the ball where and with *μ*_{1}=*μ*_{2}=*ω*^{2}, we get
3.13For example, if we choose a deformation of the disc such that , the area is preserved (at first order) and the matrix becomes diagonal with eigenvalues . This shows that, for this perturbation, (*μ*_{1}(*Ω*_{t}), *μ*_{2}(*Ω*_{t})) converge to with the tangent line . ■

To study the tangent at the point *P*_{2} corresponding to two balls with the same area, we can consider a family of domains which are the union of two balls with the same area and a small intersection of size *τ*. In the limit case , domain *Ω*_{τ} degenerates to two disjoint balls. A numerical study seems to show that this family of domains satisfies: the curve *τ*↦(*μ*_{1}(*Ω*_{τ})|*Ω*_{τ}|,*μ*_{2}(*Ω*_{τ})|*Ω*_{τ}|) has a horizontal tangent at *τ*=0. This suggests

### Conjecture 3.10

Prove that the tangent of the boundary at the point *P*_{2} corresponding to two balls is horizontal.

Finally, we calculated numerically some points on part of the boundary, which lies between the points corresponding to the ball and the two balls. Joining the points that we gathered we are able to plot region , see figure 13. All the domains that we found numerically on that part of the boundary have two axes of symmetry.

### Conjecture 3.11

Prove that the domains corresponding to points lying on the ‘free’ boundary of have two axes of symmetry. If this is the case, following the same argument as in the proof of theorem 3.9, we could prove that the set is convex in the *x*-direction.

### (b) Convex domains

In this section, we are interested in the study of the region We start presenting some numerical results obtained for polygons.

### (c) The case of polygons

In this section, we analyse some numerical results that we gathered. We calculated the first two non-trivial Neumann eigenvalues of 3000 triangles, 7800 convex quadrilaterals and 12 000 octagons. In figure 14*a* we show the results for triangles. As in the Dirichlet case, the isosceles triangles seem to be on the boundary of the region for triangles but the superequilateral and subequilateral switch roles. In the Neumann case, the superequilateral triangles define the lower part of the boundary while the subequilateral triangles define the upper boundary. We included in the figure the plot of some isosceles triangles. In figure 14*b* we plot the results for convex quadrilaterals. The larger points marked in yellow are obtained by rhomboidal domains and the continuous curve in grey corresponds to the rectangles. Our numerical results suggest that the rectangles are on the boundary of the region that we obtained for convex quadrilaterals. Actually it is a much more general property which is conjectured (see conjecture 3.14 below): the rectangles (for *L*≥2ℓ) seem to be on the boundary of .

Numerically, we observe that the lower part of the boundary (in the case of quadrilaterals) is composed of two different arcs. Denote by *T*^{r} the equilateral triangle with unit area. For , the lower boundary is defined by the superequilateral triangles and for *μ*_{1}≥*μ*_{1}(*T*^{r}) the boundary coincides with a segment of quadrilaterals, for which *μ*_{1}=*μ*_{2}, joining the points of the equilateral triangle and the square. In §3*d*, we will prove this last point. In figure 15, we plot the results obtained for convex octagons. The points corresponding to the regular polygons are marked with a plus symbol (+) and the ball is represented with a larger black point. We also plotted in blue the upper part of the boundary of region that was calculated numerically. Note that, as claimed in conjecture 3.14, rectangles seem to be on the upper boundary. We will prove a partial result in this direction in §3*d*. In both plots of figures 14 and 15, we observe that, in each case, *μ*_{1} is maximized (among polygons with a given number of sides) by the regular polygon. This result was proved in the case of triangles in Laugesen & Siudeja (2009*b*) but to our knowledge it is an open problem for the other classes of polygons. It is a similar conjecture to Polya’s problem for the Dirichlet case as mentioned in Henrot (2006, § 3.3.3). We calculated numerically the convex domain that maximizes *μ*_{2} (with unit area) and for the optimizer we obtained *μ*_{2}≈20.102. We prove in theorem 3.13 below that the maximizer is not a stadium. In figure 6, we plot a superposition of the plots of the convex domain that minimizes *λ*_{2} (dashed red line) and the convex domain which maximizes *μ*_{2} (blue line).

### (d) Some analytical results

In this section, we shall present some mathematical results for the characterization of region . As above , , , define *P*_{3}=(*μ*_{1}(*T*^{r}), *μ*_{2}(*T*^{r}))=(*μ*_{1}(*T*^{r}), *μ*_{1}(*T*^{r})) (where *T*^{r} is the equilateral triangle) and segment

### Theorem 3.12

*The set* *is not closed. It contains segment Γ*_{P1,P3}.

### Proof.

Considering a sequence of rectangles *Ω*_{n}=(0,*n*)×(0,1/*n*), we see that *μ*_{1}(*Ω*_{n})=*π*^{2}/*n*^{2} and *μ*_{2}(*Ω*_{n})=4*π*^{2}/*n*^{2}. Thus the corresponding point in converges to the origin which is not a point in ; thus this set is not closed.

For segment *Γ*_{P1,P3}, the argument is similar to those used in the proof of theorem 3.9. Let , for *t*∈[0,1] which defines a family of convex domains. As lemma 3.7 applies here, we have continuity of the Neumann eigenvalues. Thus the domains *Ω*_{t} define a continuous path connecting the points *P*_{1} and *P*_{3}, which by definition are contained in the region . Now owing to the symmetries of the domains *Ω*_{t}, we have *μ*_{1}(*Ω*_{t})=*μ*_{2}(*Ω*_{t}) (cf. lemma 3.5) and we get the conclusion. ■

We have seen in §3*a* that the maximizer of *μ*_{2} among domains with fixed area is the set composed of two identical balls. Therefore, when we add a convexity constraint, we wonder whether the maximizer would be the *convex hull* of these two balls, namely a *stadium*. The theorem below shows that this is not the case. This question was already investigated in the Dirichlet case in Henrot & Oudet (2003) and the answer was also negative. The technique of proof here is similar, though a little bit more complicated.

### Theorem 3.13

*The convex domain Ω that maximizes μ*_{2} *(among convex domains with a given area) is not a stadium (convex hull of two identical balls). More precisely its boundary does not contain any arc of circle.*

### Proof.

First of all, it is not difficult to prove the existence of a convex domain *Ω* of area 1 that maximizes the second Neumann eigenvalue (by the standard method of calculus of variations). Let us denote by *u* a (normalized) eigenfunction associated to *μ*_{2}. Let us assume, for a contradiction, that the boundary of *Ω* contains an arc of circle *γ*. We choose the centre of coordinates at the centre of the circle containing *γ*.

Using the method of proof of Henrot (2006, lemma 2.5.9) or Henrot & Oudet (2003, theorem 5), it is possible to prove that the eigenvalue *μ*_{2} is simple. Therefore, *μ*_{2} is differentiable under smooth perturbations of *γ*, and as we have a volume constraint, Hadamard formula (3.11) yields
3.14Let *θ* denote the polar angle. We also introduce the operator *A*_{θ} defined as
It is classical that *A*_{θ} commutes with the Laplacian (and also with the derivative with respect to *r*). On *γ*, we have *A*_{θ}*v*=*v*_{θ} where the subscript *θ* denotes the derivative with respect to *θ*.

Owing to the Neumann condition ∂*u*/∂*n*=0 on *γ*, we have on *γ*. Differentiating with respect to *θ*, equation (3.14) gives *u*_{θ}(*u*_{θθ}−*μ*_{2}*u*)=0. Let us prove that *u*_{θ} cannot be identically 0 on *γ* (or a part of *γ*). If this was the case, the function *v*:=*A*_{θ}*u* would satisfy
and, therefore, by the Hölmgren uniqueness theorem, we would have *v* identically 0 in a neighbourhood of *γ* and then in *Ω* by analyticity. This would imply that *u* is radially symmetric and that *Ω* is a disc, which cannot be true. Thus, since *u*_{θ} is not zero, we have
3.15We now introduce the function *w* defined by . By direct computation and equation (3.15), we have that *w* satisfies
Therefore, by the Hölmgren uniqueness theorem, the function *w* is identically 0 in a neighbourhood of *γ* and then in *Ω* by analyticity. The fact that *u* satisfies *u*_{θθ}−*μ*_{2}*u*=0 on each circle centred at the origin implies that *u* can be written
3.16Now if we plug the expression (3.16) into Δ*u*+*μ*_{2}*u*=0, we get where *J*_{1} is the usual Bessel function. Thus to get a contradiction, it suffices to prove the following property:
3.17Now if we write the boundary of *Ω* (locally) in polar coordinates *ρ*=*f*(*θ*), the property ∂*u*/∂*n*=0 reads
3.18The differential equation (3.18) can be written *f*′(*θ*)=*F*(*θ*,*f*(*θ*)) with *f*(*θ*_{0})=*R*_{0} (the radius of *γ*) with *F* a Lipschitz function. Then this differential equation has a unique solution, but it is clear that *f*(*θ*)=*R*_{0} is a solution, so equation (3.17) follows and the result is proved. ■

We recall the following conjecture already quoted in Ashbaugh & Benguria (1993).

### Conjecture 3.14

For any convex domain *Ω* prove that
3.19Equality holds if *Ω* is a rectangle with length *L* and width ℓ satisfy *L*≥2ℓ.

Here is a partial result that supports this conjecture.

### Proposition 3.15

*For any convex perturbation Ω*_{t} *of a rectangle Ω*=(0,*L*)×(0,ℓ) *with L*>2ℓ, *we have μ*_{2}(*Ω*_{t})≤4*μ*_{1}(*Ω*_{t}).

*In other words, these rectangles are local maximizers of the ratio μ*_{2}/4*μ*_{1} *among convex sets*.

### Proof.

By superposition, it suffices to consider a convex perturbation of *Ω*=(0,*L*)×(0,ℓ), which acts only on the lower side (0,*L*)×{0}. We denote by *v*(*x*)=*V* ⋅*n* the normal displacement of the point of abscissa *x*∈(0,*L*) and to preserve convexity, we assume the function *v* to be concave. Without loss of generality, we can also assume *v* to be *C*^{2}.

Let us write the first derivative (Hadamard formula) of |*Ω*|(*μ*_{2}(*Ω*)−4*μ*_{1}(*Ω*)) using equations (3.11) and (2.4), denoting by *D* this derivative, and using , we get
3.20Integrating *D* twice by parts and replacing *x* by *tL*/2*π* yields
3.21Now an elementary study of the function shows that it is always non-negative. Since *v*′′≤0, because *v* has to be chosen concave, we see that *D* is necessarily non-positive, which proves the desired result. ■

Finally, we calculated some points on the boundary of region and we show the results in figure 16. The boundary of region , already plotted in figure 13, is marked with a red dashed line.

## Acknowledgements

The work of P.A. is partially supported by FCT, Portugal, through scholarship SFRH/BPD/47595/2008, scientific projects PTDC/MAT/101007/2008, PTDC/MAT/105475/2008, FCT/CNRS and by Fundação Calouste Gulbenkian through programme *Estímulo à Investigação 2009*. The work of A.H. is partially supported by ANR GAOS.

- Received August 23, 2010.
- Accepted November 17, 2010.

- This journal is © 2010 The Royal Society