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The method of fundamental solutions applied to boundary
eigenvalue problems
Beniamin Bogosel
October 8, 2015
Abstract
We develop methods based on fundamental solutions to compute the Steklov, Wentzell and
Laplace-Beltrami eigenvalues in the context of shape optimization. In the class of smooth simply
connected two dimensional domains, the numerical method is accurate and fast. A theoretical er-
ror bound is given along with comparisons with mesh-based methods. We illustrate the use of this
method in the study of a wide class of shape optimization problems in two dimensions. We extend
the method to the computation of the Laplace-Beltrami eigenvalues on surfaces and we investigate
some spectral optimal partitioning problems.
1 Introduction
The purpose of this article is to provide some tools which facilitate the numerical study of some shape
optimization problems. Such problems consist of minimizing or maximizing a certain quantity which de-
pends on the domain geometry. The cost function which is to be optimized may depend on the geometric
properties of the domain (perimeter, area) or on some more complicated quantities given, for example,
by some partial differential equations (eigenvalues, integral energies). Finding explicitly solutions to
shape optimization problems may be difficult or even impossible in some cases. Thus, having efficient
numerical methods which can allow the study of shape optimization problems is an important issue.
This article treats the numerical optimization of functionals which depend on eigenvalue problems
defined on the boundary of the considered domain. The numerical algorithms presented in the sequel
allow the computation of the eigenvalues of the Steklov, Wentzell and Laplace-Beltrami spectra. A
precise description of each of these eigenvalues is given in the following sections. In order to compute
these eigenvalues for a given domain, we develop a method based on fundamental solutions. This type
of methods has been introduced in [24] and has been used by Antunes and Alvez in the study of various
eigenvalue problems [1],[2],[3]. Some applications to shape optimization problems are given by Antunes
and Freitas in [4] and by Osting in [27]. The advantage of such a method is the fact that there is no
need for a mesh generation at each function evaluation. This fact allows an important time economy
when thinking that it may be required that the optimization algorithm performs hundreds of function
evaluations to reach an optimizer. Another advantage is that the method based on fundamental solutions
can be very precise, and we provide a result which estimates this error in the case of the Steklov and
Wentzell eigenvalues.
The study of the Steklov spectrum is and has been a very active field of research. We cite here some
notable results concerning the optimization of these eigenvalues. Weinstock proved in [30] that the first
non-zero Steklov eigenvalue is maximized by the disk in the class of simply connected two dimensional
domains with fixed perimeter. This result was generalized in further directions by Hersch, Payne and
Schiffer in [23]. Brock proved in [10] that the first non-zero Steklov eigenvalue is maximized by the
ball in every dimension, without any constraints on the topology of the set. Other results concerning the
optimization of the Steklov spectrum under perimeter constraint were given in [17]. As underlined in [7],
all these optimization results are proved by precisely identifying the optimal shape and then proving that
the shape is indeed the desired optimizer. There are cases where the optimal shape cannot be determined
1
explicitly, and thus we have a good motivation to develop numerical tools. Concerning the existence of
the optimal shapes, some general results are given in [7] in the class of convex sets or the class of sets
which satisfy a uniform ε-cone property. The case of the Wentzell problem has been recently studied
in [14] where some conjectures were proposed concerning the shape which optimizes the first non-zero
eigenvalue. In particular, it is conjectured that the ball maximizes the first non-zero Wentzell eigenvalue.
Using our numerical method we are able to give a partial validation of this result in the two dimensional
case.
The second class of problems we study in this article concerns the partitions of a three dimensional
surface which minimize the sum of the first Laplace-Beltrami eigenvalues with Dirichlet boundary con-
ditions. Theoretical aspects concerning this problem were studied in [12],[19],[20],[21]. The numerical
study of this problem is motivated by an open question due to Bishop [6]. It is conjectured that the par-
tition of the sphere into three equal 2π/3slices has the least sum of the fundamental Laplace-Beltrami
eigenvalues. One initial objective is to test this conjecture. Previous computations were made by Elliott
and Ranner [15] by using a penalized formulation introduced in [11]. Their approach allows the study of
partitions on various three dimensional surfaces using finite element methods. Our approach is signifi-
cantly different from theirs and we believe it is of interest as it is in close connection with the computation
of the Steklov/Wentzell eigenvalues presented in the first sections of the article. The method we propose
uses techniques inspired by [8] along with a refinement procedure in the case of the sphere. The initial
part of the algorithm uses a phase field formulation where we replace each shape by an approximation of
its characteristic function. We observe that cells of the optimal partition have boundaries which are close
to being geodesics. As a result, we decide to perform a second optimization procedure in the class of
geodesic polygons. During this second phase we use a method based on fundamental solutions in order
to compute the eigenvalues of the Laplace-Beltrami operator for the cells of the partition. Our numerical
results confirm Bishop’s conjecture and show that for n∈ {3,4,6,12}it is likely that the optimal par-
tition is the one associated to the corresponding regular polyhedron. The numerical study of the cases
n= 5,n= 7 shows that the corresponding optimal partitions do not consist of geodesic polygons.
2 A numerical method for computing the Steklov/Wentzell spectrum
Let Ω⊂Rnbe an open, simply connected set with Lipschitz boundary. The Steklov eigenvalues are all
the real values σfor which the following problem has a non trivial solution.
(−∆u= 0 in Ω,
∂u
∂n =σu on ∂Ω,
It is easy to see that 0is a Steklov eigenvalue corresponding to constant functions. The Steklov eigenval-
ues form an increasing divergent sequence
0 = σ0≤σ1(Ω) ≤σ2(Ω) ≤σ3(Ω) ≤... →+∞
As usual, we can provide a variational characterization using Rayleigh quotients
σn(Ω) = inf
Sn
sup
u∈Sn\{0}RΩ|∇u|2dx
R∂Ωu2dσ , n = 1,2, ...
where Snis an ndimensional linear subspace of H1(Ω) ∩ {R∂Ωu= 0}. This variational formulations
allows us to deduce immediately the behaviour of the Steklov eigenvalues under homotheties: σk(tΩ) =
1
tσk(Ω).
Steklov eigenvalues can be computed numerically using mesh-based methods. This can be done
rather quick and in an automatic manner in FreeFem++ [18] and an example code is given in Section 6.
The mesh-based method has the disadvantage that high precision computations need a very fine mesh.
On the other hand, as meshes become more and more refined computations become slower. We present
2
below a numerical method which is fast and precise for computing the Steklov spectrum in cases where
the boundary behaves nice enough. This method can be applied to a more general class of problems.
The Steklov eigenvalue problem can be seen as a particular case of the following type of problems called
Wentzell eigenvalue problems.
(−∆u= 0 dans Ω,
−β∆τu+∂un=σu sur ∂Ω.
It is easy to see that the Steklov case corresponds to β= 0. We consider the case of star-shaped domains,
which have the advantage that their boundary can be parametrized by a radial function.
The method of fundamental solutions, introduced in [24], is a part of the class of so called mesh-free
numerical methods. The goal is to approximate the solution of a problem of the type
(Au = 0 in Ω
Bu = 0 on ∂Ω,(1)
where A, B are suitable linear differential operators. In contrast to methods using meshes, the method of
fundamental solutions considers a sufficiently rich class of functions which satisfy Au = 0 analytically in
Ω. Thus a linear combination satisfies directly Au = 0 in Ω, and the coefficients in the linear combination
are chosen such that Bu = 0 is close to zero on ∂Ω. As we will see in the following, the condition
Bu = 0 can only be imposed in a finite number of points, so the condition Bu = 0 will be satisfied
only in an approximate manner on ∂Ω. To justify our numerical approach, an error bound is provided in
Section 3, which basically says that if Bu is small enough, then uis close to the real solution. This type
of method was successfully used in [1],[3] in the study of the eigenvalues of the Dirichlet Laplacian in
two and three dimensions.
In our case, the operator Ais the Dirichlet Laplacian and the operator Bis given by −β∆τ+∂
∂n −
σId, where ∆τis the Laplace-Beltrami operator associated to ∂Ω. Our set of fundamental solutions
consists of harmonic, radial functions, with centers outside Ω. In this way, any linear combination of
such functions solves ∆u= 0. The only thing we need to do is to find the right coefficients so that the
condition −β∆τu+∂u
∂n =σu is satisfied on ∂Ω. In order to compute the expression of the Laplace-
Beltrami operator on ∂Ωwe use the formula
∆u= ∆τu+H∂u
∂n +∂2u
∂n2,
which is valid on ∂Ω. We have used the notation ∂2u
∂n2to denote (D2u.n).n. As usual, Hdenotes the
curvature of ∂Ω. For more details we refer to [22, Chapter 5].
In R2\{0}a radial solution of the Laplace equation is given by φ(x) = ln |x|. Note that this solution
has a singularity at x= 0. For every y∈R2the function ψy(x) = φ(x−y)is harmonic in R2\ {y}and
radial with center y. Given Ω⊂R2we choose y1, ..., yN∈R2\Ωand x1, ..., xN∈∂Ω. The function
x7→ α1ψy1(x) + ... +αNψyN(x)is harmonic in Ωfor every choice of the coefficients (αi)N
i=1. We
impose for i= 1...N the boundary relation
−β∆τ+∂
∂n (α1ψy1(xi) + ... +αNψyN(xi)) = σ(α1ψy1(xi) + ... +αNψyN(xi)) (2)
This amounts to solving a generalized eigenvalue problem for square matrices.
In this setting, it is straightforward to find the first eigenvalues corresponding to the generalized
eigenvalue problem determined (2), using, for example the eigs solver in Matlab. One of the main
difficulties is the choice of the points (xi)N
i=1,(yi)N
i=1. As noted in [1], an arbitrary choice for (xi),(yi)
may give fail to give a valid approximate solution for the desired eigenvalue problem. We notice the
same behaviour and we discuss below our choice of the points (xi),(yi).
3
Figure 1: Position of the source points and the evaluation points.
We tested two choices for the points (xi). The first one consists in taking a uniform division (θi)
of [0,2π]into Nintervals and then choose xi=ρ(θi)(cos θi,sin θi), where ρis the radial function
which parametrizes ∂Ω. A second choice is choosing xiat equal arclength distances on the boundary
∂Ω. We did not observe major differences between the two choices of points mentioned above. From
a computational point of view the uniform angle choice is faster. The method based on equal arclength
distances may improve the behaviour of the algorithm if the domain is thin or if the radial function has
large oscillations. Having chosen (xi), we can compute the corresponding outer normals (~ni)and we
define yi=xi+ 0.1·~ni. It seems that the choice of the factor 0.1is essential in our setting. Even slight
perturbations of this factor give results which are far from the actual Steklov eigenvalues of Ω. This is
due to the fact that for larger or smaller values of this parameter, the matrices involved in the computation
are ill conditioned.
It is not hard to see the limitations of this method of fundamental solutions. Since linear combinations
of fundamental solutions have singularities at their source points, it is clear that these source points must
be located outside the computation domain. Another aspect is that source points must be distinct so that
equations (2) do not repeat themselves. These two aspects already suggest that domains with cusps or
re-entrant parts are not covered by our algorithm.
3 Error estimates
In the case of the Dirichlet Laplacian, the result proved by Moler and Payne in [25], states that if a
function usatisfies −∆u=λu in Ωand uis sufficiently small on ∂Ωthen λis close to an eigenvalue
of the Dirichlet-Laplace operator associated to Ω. In order to validate our numerical computations, we
provide a similar result below, in the case of the Steklov eigenvalue problem. In the following paragraphs
we assume that Ωhas Lipschitz boundary and that it has finite perimeter. In the following we denote
V(Ω) = {u∈L2(∂Ω) : R∂Ωu= 0}.
As in [14] we introduce the Hilbert space H(Ω) = {u∈H1(Ω) : Tr(u)∈H1(∂Ω),R∂Ωu= 0}
where Tr is the trace operator. In the case β= 0 it suffices to take H(Ω) = H1(Ω). Consider for
f∈V(Ω) the minimization problem
min
u∈H(Ω)
1
2ZΩ
|∇u|2+βZ∂Ω
|∇τu|2−Z∂Ω
uf
which has a unique solution. This solution satisfies the weak form
ZΩ
∇u· ∇ϕ+βZ∂Ω
∇τu∇τϕ=Z∂Ω
fϕ, ∀ϕ∈C1(Ω),(3)
of the partial differential equation
(−∆u= 0 in Ω
−β∆τu+∂u
∂n =fon ∂Ω.,(4)
4
where ∆τis the Laplace-Beltrami operator and ∇τis the tangential gradient associated to ∂Ω. Thus, we
can define the resolvent operator Rβ:L2(∂Ω) →H(Ω) associated to this equation. The trace operator
T:H(Ω) →V(∂Ω) being continuous it follows that the operator T◦Rβ:V(Ω) →V(Ω) is compact
and injective. We can define its inverse Aβ:D(Aβ)⊂V(Ω) →V(Ω). Since T◦Rβis a compact op-
erator, the spectrum of the operator Aβconsists of an increasing sequence of eigenvalues λk,β (Ω) which
diverges. The corresponding eigenfunctions form a Hilbert basis for V(Ω). By considering the constant
function 1associated to the zero eigenvalue of this operator, we can say that the set of eigenvalues forms
a Hilbert basis of L2(∂Ω). The following result proves that the operator T◦Rβis bounded and gives an
idea of how to find its norm. By abuse of notation we will denote the trace of a function w∈H1(Ω) by
w.
Proposition 3.1. Let Ωbe a bounded, open domain with Lipschitz boundary. Suppose f∈V(Ω) and
w=Rβf∈H1(Ω). Then there exists a constant C, depending only on Ω, such that
kwkL2(∂Ω) ≤CkfkL2(∂Ω).
Proof: The trace inequality (Chapter 4.3 [16]) for Ωimplies the existence of a constant C1(depending
only on Ω) such that kukL2(∂Ω) ≤C1kukH1(Ω) for every u∈H1(Ω). The Poincare-Wirtinger inequality
implies the existence of a constant C2which depends only on Ωsuch that k˜wkL2(Ω) ≤C2k∇wkL2(Ω),
where ˜w=w−1
|Ω|kwkL2(Ω). The weak formulation of the equation Rβf=wand the Cauchy-Schwarz
inequality imply that
ZΩ
|∇ ˜w|2+βZ∂Ω
|∇τ˜w|2=Z∂Ω
f˜w≤ kfkL2(∂Ω)k˜wkL2(∂Ω).
Using the remarks above, we obtain
k˜wk2
L2(∂Ω) ≤C2
1(k˜wk2
L2(Ω) +k∇ ˜wk2
L2(Ω))≤C2
1(1 + C2
2)k∇ ˜wk2
L2(Ω).
Thus
k˜wk2
L2(∂Ω) ≤C2
1(1 + C2
2)kfkL2(∂Ω)k˜wkL2(∂Ω) ,
which implies
k˜wkL2(∂Ω) ≤C2
1(1 + C2
2)kfkL2(∂Ω).
On the other hand, since whas average 0on ∂Ω, we know that the L2(∂Ω) norm of w+cis minimal
when c= 0 (here cis a constant). Therefore
kwkL2(∂Ω) ≤ k ˜wkL2(∂Ω) ≤C2
1(1 + C2
2)kfkL2(∂Ω).
The constants C1, C2can be expressed explicitly in terms of the domain Ω. The constant C1depends
on the Lipschitz constant of Ωand C2= 1/λ1(Ω), where λ1(Ω) is the first eigenvalue of the Dirichlet
Laplacian operator on Ω. These quantities may be evaluated on particular regions, and together with the
result presented below can provide an exact error estimate.
Using ideas similar to the ones used by Moler and Payne in [25], we are able to prove the following
error estimate. For simplicity of notation we omit the reference to βfrom Rβ. In the sequel, for the sake
of simplicity of exposition, we use the same notation for a function and for its trace.
Theorem 3.2. Consider Ωa bounded, open domain with Lipschitz boundary and finite perimeter. Sup-
pose that uεsatisfies the following approximate eigenvalue problem:
−∆uε= 0 in Ω
−β∆τuε+∂uε
∂n =λεuε+fεon ∂Ω.(5)
5
Denote wε=Rfε. Let δ=kwεkL2(∂Ω)
kuεkL2(∂Ω)
and suppose that δ < 1. Then there exists an Wentzell
eigenvalue λksatisfying
|λε−λk|
λk
≤δ.
Proof: We know that there exists a Hilbert basis of L2(∂Ω) formed of Wentzell eigenfunctions (un)
corresponding to the Wentzell eigenvalues λnof Ω. We denote the standard scalar product in L2(∂Ω)
by (u, v) = Z∂Ω
uv. Let an= (uε, un), bn= (wε, un). We know that R(λεuε+fε) = uεand
Run=un/λn. The resolvent operator Ris symmetric, thus
an= (uε, un) = (R(λεuε+fε), un)
= (λεuε+fε, Run)
=1
λn
(λεuε+fε, un)
=1
λn
(λεan+λnbn).
Thus, for every nwe have λn−λε
λn
=bn
an
. Since (λn)is increasing and divergent, there exists an index
ksuch that |λk−λε|
|λk|= min
n
|λn−λε|
|λn|.
For this index kwe have |λk−λε|
|λk||an| ≤ |bn|,
for all nand
|λk−λε|2
|λk|2
∞
X
n=1
a2
n≤
∞
X
n=1
b2
n.
This is exactly
|λk−λε|
|λk|≤δ,
which finishes the proof.
The only hypothesis in the above theorem which needs to be verified in order to apply it in our case
is that we can solve the partial differential equation wε=Rβfεin the case where fεis a combination of
the fundamental solutions. As we have seen the necessary and sufficient condition is R∂Ωfε= 0. Note
that this condition can always be satisfied by adding a constant function to the family of fundamental
solutions.
4 Testing the numerical method
Let’s note that the first Wentzell eigenvalue of Ωis λ0,β(Ω) = 0, corresponding to a constant eigenfunc-
tion. We denote λk,β (Ω) the k-th Wentzell eigenvalue after λ0,β (Ω). There are few shapes for which the
Wentzell spectrum (or the Steklov spectrum in the case β= 0) is known analytically. One such shape is
the unit disk D1, which has the eigenvalues
λk,β (D1) = k+ 1
2+βk+ 1
22
.
As an initial test for our algorithm we computed the Wentzell spectrum of the disk for N= 300 points
on ∂D1and 300 corresponding fundamental solutions. For β= 0 we have 10 digits of precision for the
6
0 20 40 60 80 100
0
0.5
1
1.5
2
2.5 x 10−7
β
error
Absolute errors − unit disk
Figure 2: Absolute errors for β∈[0,100] - the case of the disk
our algorithm FreeFem++ (refined meshes)
k MFS 19146 N53236 N211290N474634N
1 0.712751 0.712989 0.712837 0.712773 0.712761
2 0.940247 0.940538 0.940352 0.940274 0.940259
3 1.381278 1.38211 1.38158 1.38135 1.38131
4 1.443204 1.44411 1.44353 1.44329 1.44324
5 3.146037 3.14712 3.14643 3.14614 3.14608
6 3.443637 3.44496 3.44411 3.44376 3.44369
7 3.757833 3.761 3.75897 3.75812 3.75796
8 3.922821 3.9263 3.92407 3.92313 3.92296
9 4.274362 4.28034 4.27651 4.2749 4.2746
10 4.693206 4.70035 4.69578 4.69385 4.6935
Table 1: Comparison with FreeFem++, β= 0 (Steklov) for the shape given in Figure 1
first 10 lowest eigenvalues. In Figure 2 we plot the absolute error for the first 10 Wentzell eigenvalues
β∈[0,100]. We note that for β= 100 we still have 6digits of precision.
In order to test our algorithm for shapes for which no analytical expression is known for the Wentzell
eigenvalues, we use FreeFem++ [18], which uses mesh-based methods. The tests we performed show
that our results are in good correspondence with the values found with FreeFem++. The downside of
the mesh-based method is the execution time, which is significantly more important when the number
of mesh triangles is high. An example of implementation is presented in Section 6. In Tables 1, 2,
3 we compare the Wentzell eigenvalues computed with our method (MFS) and the ones obtained with
FreeFem++. As a test case we take the shape found in Figure 1, for various values of β. Note that
as the number of triangles increases, the values computed with the FreeFem++ method approach the
values found with our algorithm. We underline the fact that our algorithm runs in approximately 0.1
seconds1, whereas the FreeFem++ algorithm, with over 450000 triangles takes about a minute on the
same machine.
Another way of testing our algorithm is to do numerical optimization procedures for shape optimiza-
tion problems with known optimizers. There are many such results for the case β= 0 (the Steklov
eigenvalue problem). We start from a random shape and look if the algorithm converges to the expected
shape, which is usually the disk. We mention that all computations are made in the class of simply
connected sets. We were able to test our algorithm in the following cases:
1Machine configuration: 2.2Ghz quad-core i7 processor, 6Gb RAM memory
7
our algorithm FreeFem++ (refined meshes)
k MFS 19146 N53236 N211290N474634N
1 2.375744 2.37628 2.37594 2.37579 2.37577
2 2.644741 2.6453 2.64494 2.64479 2.64476
3 8.042223 8.04527 8.04332 8.0425 8.04234
4 8.257585 8.26043 8.25861 8.25784 8.2577
5 16.909967 16.9197 16.9135 16.9108 16.9104
6 17.383930 17.3932 17.3873 17.3848 17.3843
7 28.883924 28.9094 28.8931 28.8862 28.8849
8 29.113307 29.1374 29.122 29.1155 29.1143
9 43.718607 43.77 43.7371 43.7232 43.7207
10 44.142742 44.1996 44.1632 44.1479 44.145
Table 2: Comparison with FreeFem++, β= 2 for the shape given in Figure 1
our algorithm FreeFem++ (refined meshes)
k MFS 19146 N53236 N211290N474634N
1 4.750048 4.75121 4.75047 4.75015 4.75009
2 5.02106 5.02224 5.02148 5.02117 5.02111
3 17.557103 17.5638 17.5595 17.5577 17.5574
4 17.774667 17.781 17.777 17.7752 17.7749
5 38.179237 38.2016 38.1873 38.1812 38.1801
6 38.65575 38.6771 38.6634 38.6577 38.6566
7 66.764114 66.8228 66.7852 66.7694 66.7665
8 66.995238 67.0507 67.0152 67.0002 66.9975
9 102.91875 103.038 102.962 102.929 102.924
10 103.34252 103.474 103.39 103.354 103.348
Table 3: Comparison with FreeFem++, β= 5 for the shape given in Figure 1
8
•max σ1(Ω) is achieved when Ωis a disk, in the case of perimeter and area constraints ([30],[10]);
•max σ1(Ω)σ2(Ω) is achieved when Ωis a disk, in the case of perimeter and area constraints ([23]);
•min
n
X
k=1
1
σk(Ω) is achieved when Ωis a disk, in the case of perimeter and area constraints [23]);
•max σk(Ω) under a rotational symmetry of order qis achieved by a disk in the case of the perimeter
constraint ([5]).
This method of fundamental solutions can be adapted to compute the Laplace-Beltrami spectrum of
a closed simple curve in kR2. We can consider solving the equation
∆τ(α1ψy1(xi) + ... +αNψyN(xi)) = λ(α1ψy1(xi) + ... +αNψyN(xi)), i = 1...N (6)
which also leads to a generalized eigenvalue problem. The Laplace-Beltrami spectrum of a one dimen-
sional curve depends only on its length and is given by λk=k+1
222π
L2. The method of fundamental
solutions computes these values with a relative error of order 10−7(with the same parameters: 300
boundary points and exterior points at distance 0.1of the boundary). This good behaviour motivates the
extension of the method to the computation of Laplace-Beltrami eigenvalues for subsets of the the unit
sphere, presented in Section 7.
We may use Theorem 3.2 in order to have a quantitative evaluation of the error on a general do-
main. The result cited above states that the relative error made in the numerical computations is of order
kfεkL2(∂Ω), where fεis the error term in (5). We may estimate numerically fεas follows: given a shape
Ω, we compute its Steklov/Wentzell eigenvalues with the algorithm presented in previous sections. We
know that the eigenvalue equation is satisfied to machine precision on the discretization points chosen on
∂Ω. In order have a more precise evaluation of what happens between these points we make a refinement
by placing 100 supplementary points between every two discretization points. We evaluate fεin each of
these points. The maximal value found for fεgives us an estimate of the general error. Below you can
see plots of fεfor the first 10 eigenvalues in three different cases. By looking at the maximal errors, we
can observe that kfεkL2(∂(Ω)) is of order 10−6or smaller. As expected, different domains give different
behaviours and the precision can be much higher in some particular cases.
5 Numerical optimization of functionals depending on the Wentzel spec-
trum
Using the algorithm presented in the previous sections, we can study numerically shape optimization
problems regarding the Wentzell spectrum in the particular case of star-shaped domains. We consider
domains parametrized by their radial functions ρ: [0,2π)→R+. We approximate ρby the truncation
of its Fourier series to 2n+ 1 coefficients:
ρ(θ)≈a0+
n
X
i=1
aicos(iθ) +
n
X
i=1
bisin(iθ).
In this way, we have an approximation of σk(Ω) using a finite number of parameters. Using the shape
derivative formula provided in [14, Section E] we can deduce that
∂σk
∂ai
=Z2π
0|∇τuk|2− |∂nuk|2−λH|uk|2+β(HI−2D2b)∇τuk.∇τukρ(θ) cos(iθ)dθ
and
∂σk
∂bi
=Z2π
0|∇τuk|2− |∂nuk|2−λH|uk|2+β(HI−2D2b)∇τuk.∇τukρ(θ) sin(iθ)dθ
9
0 1 2 3 4 5 6
−4
−3
−2
−1
0
1
2
3
4x 10−12
θ
fε
Error for β = 0
error 2
error 4
error 5
0 1 2 3 4 5 6
−3
−2
−1
0
1
2
3x 10−7
θ
fε
Error for β = 1
error 2
error 4
error 5
0 1 2 3 4 5 6
−3
−2
−1
0
1
2
3x 10−7
θ
fε
Error for β = 2
error 2
error 4
error 5
Figure 3: Graph of the error term in the computation of the Steklov and Wentzell eigenvalues of indexes
k= 2,4,5(various values of β).
We use the notation Hfor the mean curvature of ∂Ω. We denote by D2bthe hessian of the signed distance
function, or equivalently, the differential of the normal vector. We have denoted ukthe eigenfunction
corresponding to σk(Ω) normalized in L2(∂Ω).
Since we can approximate σk(Ω) by a function σk(a0, a1, ..., an, b1, ..., bn)for which we know the
gradient with respect to every component, we can use a gradient descent approach for solving different
optimization problems related to the Steklov eigenvalues. This approach was used in [27] and [4] for
optimizing functionals of the eigenvalues of the Dirichlet Laplacian.
In the recent article of Dambrine, Lamboley and Kateb [14], the authors prove that the ball is a
local minimizer for the first non-zero Wentzel eigenvalue if β≥0, under volume constraint. Using the
fact that λ1,β(BR)is decreasing with respect to R(we denote BRthe ball of radius R), we can deduce
that if the ball is a maximizer for the perimeter constraint, then it is also a maximizer for the volume
constraint. It is a well known fact, due to Weinstock [30] and Brock [10], that when β= 0 the ball
is the optimizer for both volume and perimeter constraints. Using our algorithm, we searched for the
shape which optimizes λ1,β(Ω) in two dimensions. For both perimeter and area constraints we obtained
that the disk is the numerical maximizer of λ1,β among two dimensional simply connected shapes. We
performed tests for β∈[0,100], but we believe it to be true for every β > 0since for large values of
β,λ1,β(Ω)/β converges to the first Laplace-Beltrami eigenvalue of ∂Ω. We also performed tests in the
case of the area constraint for k= 2,3,4,5and we present the results in Table 4.
We present some conjectures verified numerically using our algorithm. Many of them are related to
results known to be true in the Steklov case (β= 0), namely, the results of Hersch, Payne and Schiffer
[23]. All these results are for domains which are simply connected and star-shaped.
•max λ1,β(Ω) is acheived by the disk;
•min
n
X
k=1
1
λk,β (Ω) is achieved by the disk;
•We say that A⊂ {0,1,2,3, ...}has the property (P)if 1∈Aand 2k∈A⇒2k−1∈A. If A
10
λ1λ2λ3λ4λ5
β= 0
(Steklov)
λ1= 1 λ2= 1.64 λ3= 2.33 λ4= 2.97 λ5= 3.66
β= 0.1
λ1= 1.1λ2= 1.80 λ3= 2.65 λ4= 3.42 λ5= 4.3
β= 0.5
λ1= 1.5λ2= 2.39 λ3= 4 λ4= 4.53 λ5= 7.5
β= 100
(large)
λ1= 101 λ2= 101 λ3= 402 λ4= 402 λ5= 903
Table 4: Numerical maximizers for the first five Wentzell eigenvalues for different values of β. The areas
of the domains are equal to π
has the property (P)then X
k∈A
1
λk,β (Ω) is minimized by the disk in the case of a area and perimeter
constraint. For example 1
λ1,β(Ω) +1
λ3,β(Ω)
+1
λ4,β(Ω) is minimized by the disk in the case of the
area constraint and the perimeter constraint. This was verified for various sets Awith property (P)
with A⊂ {0,1, ..., 15}.
As underlined before, in the Steklov case with a perimeter constraint, the simple connectedness is
essential. Making a small hole in the center of the disk and rescaling in order to have the same perimeter
increases the first eigenvalue. This behaviour can be seen in Figure 4 in some computations made with
FreeFem++. As proved by Brock [10], if we impose an area constraint then the simple connectedness
condition is not necessary. We may ask if this is the case for the Wentzell eigenvalues. The answer is
negative, as can be seen in Figure 5 for β= 0.1. Making a small hole and rescaling to have the same
area increases the first Wentzell eigenvalue of a disk.
6 The FreeFem++ code for solving the Wentzell eigenvalue problem
int i;
real t,beta = 2;
// Domain definition using a radial function
border C(t=0,2*pi){x=cos(t)*(1+0.1*cos(t)+0.1*cos(5*t)+
0.1*sin(2*t)-0.1*sin(5*t));
y=sin(t)*(1+0.1*cos(t)+0.1*cos(5*t)+
11
Figure 4: Behaviour of the Steklov eigenvalue when making holes. The images represent a unit disk
with holes of radii 0.03 and 0.04, rescaled to have total perimeter 2π. Note that the corresponding first
eigenvalues are higher than 1which is the first eigenvalue of a disk of perimeter 2π.
Figure 5: Behaviour of the Wentzell eigenvalue when making holes. The images represent two disks with
holes in their centers rescaled to area π. The first eigenvalues are higher that 1.1which is the Wentzell
eigenvalue of a disk for β= 0.1.
12
0.1*sin(2*t)-0.1*sin(5*t));}
mesh Th = buildmesh (C(500));
fespace Vh(Th,P1); // Build P1 finite element space
Vh uh,vh;
// Define the problem via weak formulation
varf va(uh, vh) = int2d(Th)( dx(uh)*dx(vh)+dy(uh)*dy(vh))+
int1d(Th,1)(beta*(dx(uh)*dx(vh)-
dx(uh)*N.x*(N.x*dx(vh)+N.y*dy(vh))-
dx(vh)*N.x*(N.x*dx(uh)+N.y*dy(uh))+
N.x*(dx(vh)*N.x+dy(vh)*N.y)*N.x*(dx(uh)*N.x+dy(uh)*N.y)+
dy(uh)*dy(vh)-
dy(uh)*N.y*(dx(vh)*N.x+dy(vh)*N.y)-
dy(vh)*N.y*(dx(uh)*N.x+dy(uh)*N.y)+
(N.y)ˆ2*(dx(vh)*N.x+dy(vh)*N.y)*(dx(uh)*N.x+dy(uh)*N.y)));
varf vb(uh, vh) = int1d(Th,1)(uh *vh);
// Solve the generalized eigenvalue problem
matrix A = va(Vh, Vh ,solver = sparsesolver);
matrix B = vb(Vh, Vh);
real cpu=clock(); // get the clock
int eigCount = 10; // Get first Eigenvalues
real[int] ev(eigCount); // Holds Eigenfunctions
Vh[int] eV(eigCount); // Holds Eigenfunctions
// Solve Ax=l*Bx
int numEigs = EigenValue(A,B,sym=true,sigma=0,
value=ev,vector=eV);
for(int i=0;i<eigCount;i++) // Plot the spectrum
plot(eV[i],fill=true,value=true,cmm= ev[i]);
cout << " CPU time = " << clock()-cpu << endl;
for(i = 0;i<eigCount;i++)
cout << ev[i] << endl;
7 Laplace-Beltrami eigenvalues on the sphere
Motivated by the fact that the Laplace-Beltrami eigenvalues of a closed curve in R2can be found using
fundamental solutions we extend the method to the case of the unit sphere in R3. In order to do this we
consider the extended problem
(−∆τu=λu on S2
−∆u= 0 in a neighborhood of S2.(7)
The motivation behind this consideration is the following decomposition of the Laplacian
∆u= ∆τu+H∂u
∂n +∂2u
∂n2.(8)
For a proof of (8) and more details we refer to [22]. As usual, Hdenotes the mean curvature of the
surface. We denote ∂2u
∂n2= (D2n.u).u.
As before, we seek uas a linear combination of radial harmonic functions in R3which do not have
singularities on S2.
We consider the fundamental solution of the Laplace equation in three dimensions given by φ(x) =
1/|x|. We choose a family of Nevaluation points (xi)on S2which are uniformly distributed. We can do
an explicit construction starting from a dodecahedron in the case of the sphere, or we can use DistMesh
13
Figure 6: Absolute errors - approximation of the first 10 Laplace-Beltrami eigenvalues of S2.
[26] in general situations. The source points (yi)are chosen on the normals at S2in xiat a fixed distance
r. As we will see below, the behaviour of the error depends on rand N. These parameters must be chosen
such that the matrices involved in the computations are well conditioned, in order to have meaningful
results. For each yiwe consider the fundamental solution centered in yidefined by ψi(x) = φ(|x−yi|).
We seek uin the form
u=α1ψy1+... +αNψyN.
We impose the eigenvalue condition in each of the points (xi)and we obtain the equations
−∆τ(α1ψy1(xi) + ... +αNψyN(xi)) = λLB(S2)(α1ψy1(xi) + ... +αNψyN(xi)), i = 1...N. (9)
Solving this generalized eigenvalues problem we expect to find the values of the Laplace Beltrami eigen-
values on the unit sphere. The explicit eigenvalues are of the form ℓ(ℓ+ 1) with multiplicity 2ℓ+ 1, with
ℓ≥0. We recall that ris the distance from the exterior points (yi)to the boundary of the sphere. The
choice of the sample points (xi)is not random. As noted in [3], the sample points should be distributed
evenly across the surface in order to obtain meaningful results. We tried multiple choices for the points
(xi):
•Uniform sphere mesh found with Distmesh [26].
•The layer method described in [3]
•A uniformly refined mesh of the sphere starting from an icosahedron.
For all these choices of points we observe that the values obtained with our algorithm are very close to
the analytical ones. An analysis of the dependence of the absolute error of the parameter rand on the
number of sample points is given in Figure 6. We can see that the error decreases drastically with r.
We also observe that when we have a large number of points and large rthe computation is not stable
anymore. These estimates are valid for the first 10 eigenvalues.
We can use the method of fundamental solutions in order to compute the Laplace-Beltrami eigenval-
ues with Dirichlet boundary conditions of a shape ω⊂S2. In order to do this we consider only sample
points xi∈ωand approximate λLB(ω)using a variation of equation (9). Indeed, let (xi)N−M
i=1 be points
in the interior of ω(relative to S) and (zi)M
i=1 be points on ∂ω (relative to S). Using the same method of
fundamental solutions, the eigenvalue condition is exactly (9). The boundary conditions can be written
as
α1ψy1(zj) + ... +αNψyN(zj) = 0, j = 1...M. (10)
It is possible to couple the systems (9) and (10) into one single generalized eigenvalue problem in the
form A
Bv=λX
Ov(11)
where
14
Figure 7: Behavior of the L(2π/3) eigenvalue approximation with respect to the parameter r, for 217
and 817 sample points
•A= (−∆τψyj(xi)), i = 1...N −M, j = 1...N
•B= (ψyj(zk)), k = 1...M, j = 1...N
•X= (ψyj(xi)), i = 1...N −M, j = 1...N
•Ois the zero matrix of size (N−M)×N.
•v= (α1, ..., αN)T.
The points (xi),(zj)are chosen by performing a triangulation of the set ω⊂S, which in our computa-
tions will always be a geodesic polygon. In order to compute such a triangulation, we divide the polygon
in to triangles and then refine this triangulation multiple times by considering the classical midpoint
refinement.
In order to test our computational method, we consider some particular subsets of the sphere for
which some of the eigenvalues are known explicitly. In the following we call lens of angle θ, a portion
of the sphere contained between two half-meridians which make angle θ. We denote the first eigenvalue
of a lens of angle θby L(θ). We call a double-right triangle of angle θa half (divided by the ecuatorial
circle) of a lens of angle θ. We denote the first eigenvalue of a double-right-triangle of angle θby R(θ).
The following analytical values are known for R(θ), L(θ):
•L(θ) = π
θπ
θ+ 1(see [29]) - numerical example in Figure 7.
•R(π/3) = 20, R(π/2) = 12, R(π) = 6 (see [29]) - numerical examples in Figure 8)
Another interesting spherical triangle is the one which realizes the partition of the sphere into 4con-
gruent equilateral triangles. We denote one such triangle by T. The computation of the first eigenvalue of
this triangle came up in [28] in the study of the expected capture time of some brownian motion predators
on the line. The numerical value computed in the above reference is λLB
1(T) = 5.1589 (represented by
the green line in Figure 9). We compute numerically its first eigenvalue and compare it to the values
presented in the cited article (see Figure 9). We observe that for r∈[1.8,1.9] the error made by our
algorithm is really small. We see again the instability in the computation as rincreases. In order to fur-
ther test this numerical value, we used a finite element discretization of the triangle T, and we compute
the first eigenvalue in terms of on a mesh having 98000 points. We obtain λLB
1(Tfem) = 5.1593, which
is close to both the result of [28] and our values. We note, though, that in order to reach this precision,
more than 50 times more points are needed in the discretization.
Until now we only considered exact subsets of S2. We can extend our method to compute the spec-
trum associated to an approximation ϕof χω. In order to do this, we use the relaxed formulation inspired
from [13], [8] given by
−∆τu+µu =λu,
15
Figure 8: Behavior of the approximation of R(π/3) (left) and R(π/2) (right) with respect to r
Figure 9: Behavior of the approximate first eigenvalue of Twith respect to r(left), corresponding first
eigenfunction (right)
16
where µis a capacitary measure which penalizes points outside ω. This relaxed formulation includes the
classical case. We can compute the eigenvalues of ω⊂S2by imposing µ= +∞in S2\ωand µ= 0 in
ω. The advantage is that we work on the whole sphere and the measure µtakes into account the change
of shape. Using this technique, it is possible to study the partitions of the sphere which minimize the
sum of their first Laplace-Beltrami eigenvalues. The Euclidean case of this problem was considered by
Bourdin, Bucur and Oudet in [8], while the spherical case was recently treated by Elliott and Ranner in
[15] using a different method.
We choose µ=C(1 −ϕ)dσ and the penalized formulation becomes
−∆τu+C(1 −ϕ)u=λu. (12)
This can be written in matrix form as
(A+Cdiag(1 −ϕ)B)v=λBv,
where
•A= (−∆τψj(xi)), i, j = 1...N
•B= (ψj(xi)), i, j = 1...N
•v= (α1, ..., αN)T
•diag(1 −ϕ)is the diagonal matrix with diagonal entries 1−ϕ.
For the generalized eigenvalues computations we use the Matlab eigs function. In order to be able to
perform an optimization, we need to compute the gradient of the eigenvalue with respect to ϕ. For this
we have two options:
•Compute the gradient in the analytic setting and obtain ∇λ(ϕ) = −Cv2where vis the associated
eigenfunction. This was proved in [8].
•Compute the gradient in the discrete setting, by differentiating the generalized eigenvalue problem.
In order to do this, we need the corresponding right eigenvector vand the left eigenvector w. We
obtain that
∇λ(ϕ) = −Cw ⊗Bv/(wTBv),
where ⊗is the usual tensor product.
Both of the above methods work, but the second needs to perform two times the amount of computations
as the first, since we need both the left and right corresponding eigenvectors. In our computations we
prefer the first approach, as it is faster. The optimization is made using a standard gradient descent
algorithm. We need to impose the partition condition at each iteration, and we do this by applying the
following projection operator
Π(ϕl) = |ϕl|
Pn
i=1 |ϕi|.
8 Numerical optimal partitions
There is an interest in computing numerically the spectral optimal partitions on the sphere. This interest
is motivated by the fact that problems that are simple to state regarding these optimal partitions are still
open. Bishop proved that the partition of S2into two parts ω1, ω2which minimizes λLB
1(ω1) + λLB
1(ω2)
consists of two half-spheres. The similar problem of finding the minimizer of
λLB
1(ω1) + λLB
1(ω2) + λLB
1(ω3),(ω1, ω2, ω3)partition of S2,
17
Figure 10: The optimal configuration for n= 8 (left) and n= 5(right). The black lines are geodesic arcs
connecting the vertices of a face.
is still open. In the same article [6] it is conejectured that the optimal partition in the case n= 3 is made
of three 2π/3-lens. A similar problem, which is a consequence of Bishop’s conjecture, was treated by
Helffer et al. in [21]. They proved that the partition of the sphere into three 2π/3-lens minimizes the
quantity
max
i=1,2,3λLB
1(ωi),(ω1, ω2, ω3)partition of S2.
Initial numerical computations of optimal spectral partitions on S2were computed by Elliott and Ran-
ner in [15]. They confirmed numerically Bishop’s conjecture, and they made computations for n=
3,4,5,6,7,8,16,32. Their method is based on a penalized energy formulation of the partitioning prob-
lem introduced in [11].
In the following, we propose a different approach, inspired by the two dimensional case studied
by Bourdin, Bucur and Oudet [8]. We represent each phase ωiof the partition by a density function
ϕi:S2→[0,1] . The partition condition then translates to Pn
i=1 ϕi= 1. Given ϕ, a density function
approximating ω, we consider the problem
−∆τu+C(1 −ϕ)u=λLB
1(C, ϕ)uon S2(13)
with C >> 1. As in [8], it can be proved that the mapping ϕ7→ λLB
1(C, ϕ)is concave and as C→ ∞
we have λLB
1(C, χω)→λLB
1(ω).
We were able to compute numerically the optimal partitions for
n
X
i=1
λLB
1(ωi),(ω1, ..., ωn)partition of S2,
for n∈[3,24] ∪ {32}, using about 5000 sample points. It is interesting to note that for n∈ {3,4,6,12}
we obtain the regular tiling of the sphere. For n∈ {5,7,8,32}we obtain the same results as Elliott and
Ranner. For n= 16 we obtain something slightly different: they obtained a configuration of 4equal
hexagons, 4equal pentagons and another 8equal pentagons. We obtain 4equal hexagons and 12 equal
pentagons, which is plausible, since this is the most regular 16 tiling of the sphere.
In [15] it is conjectured that the common boundary of two adjacent cells is a geodesic arc. This
fact can be can be seen for he case n= 8 in Figure 10, where we plotted some geodesic arcs on top of
the results obtained using density functions. In our density results we can observe that for n≥8the
common boundary of any two cells is really close to being a geodesic arc. This motivated us to search
for the optimal partitions among geodesic polygons which is a problem depending only on a low number
of parameters. We discuss later the fact that even if for large nthe optimal partition cells are close to
being geodesic polygons, this is not true for n∈ {5,7}.
A first step in the refined optimization procedure in the class of geodesic polygons is to extract
the topological structure from the optimal densities. For each polygon in the partition we compute the
18
corresponding first eigenvalue using the method presented in equation (11). In order to optimize the
position of the vertices it is possible to write derivatives with respect to each coordinate of the vertices.
Instead of doing this we use a simpler algorithm, which avoids the computation of numerical integrals
on the surface of the sphere. We use the following discrete algorithm with a probabilistic touch:
•For each point Pconsider a family of qtangential directions (vi)q
i=1 chosen as follows: the first
direction is chosen randomly and the rest are chosen so that the angles between consecutive direc-
tions are 2π/q.
•Evaluate the cost function for the new partition obtained by perturbing the point Pin each of the
directions viaccording to a parameter ε.
•Choose the direction which has the largest decrease, and this is the updated partition.
•If there is no decrease for each of the points of the partition, then decrease ε.
This algorithm converges and its accuracy has been tested by choosing different starting configura-
tions. We observe that it always converges to the same partition. The optimal densities and the opti-
mal partitions consisting of geodesic polygons are presented in the following figures. We present the
results obtained in the cases corresponding to n∈ {3,4, ..., 16,20,32}. Computations were made for
n∈ {17,18,19,21,22,23,24}, but the presentation of the corresponding figures does not present any
particular interest. We remark the fact that for n≥14 optimal partitions seem to consist of 12 pentagons
and n−12 hexagons.
We observe that for n∈ {4,6,12}the optimal partition cells are regular geodesic polygons cor-
responding to the tetrahedron, the cube and the dodecahedron. For nlarge enough, the partition cells
become so close to geodesic polygons that we cannot visualize the difference. However, the case n= 5,
seen in Figure 10, raises some questions about the validity of the claim that boundaries are geodesics.
We have devised a numerical test of this claim which is presented below.
•take the optimal partition into geodesic polygons and add supplementary variable points at the
midpoint of every edge;
•perform again the optimization in this new setting.
To illustrate this better, we give more details concerning the case n= 5. The optimal partition into
geodesic polygons consists of two triangles and four rectangles. Adding the midpoints as variables gives
us a new configuration of two hexagons and three octagons. What we observed in the cases n= 5, n = 7
is that adding vertices at midpoints we get a new optimal partition which has a slightly lower value of the
cost function. We believe that this decrease in the cost function allows us to conclude that the optimal
cells do not always consist of geodesic polygons. The test cases n= 5 and n= 7 mentioned above is
presented in Figure 11. For n= 5 we obtain a decrease of 0.04 in the cost function, while for n= 7 we
obtain a decrease of 0.02.
It is possible to perform similar computations for surfaces which are more complex than the sphere.
The examples we considered are similar to the ones considered in the perimeter case: a torus with outer
radius R= 1 and the inner radius r= 0.6and the Banchoff-Chmutov surface of order 4. Using
the proposed algorithm, we can deal with any surface, as soon as we have a qualitative triangulation.
On more complex surfaces only the optimization algorithm using density functions is used, since the
boundaries of the cells cannot be easily characterized.
An equally interesting problem is finding the partition which minimizes the quantity maxiλLB
1(ωi).
Theoretical aspects of the problem as well as a complete analysis of the case n= 3 were given in
[21]. It is known that if the solution of the problem corresponding to the sum consists of cells with
the same eigenvalue, then this is also a solution of the maximum problem. In our computations, only
the regular partitions corresponding to n∈ {3,4,6,12}have this property, and thus they are solutions
for the maximum problem as well. In all remaining cases we obtained at least two cells with different
eigenvalues, which means that our partitions are not optimal for the maximum. Optimizing the maximum
19
Figure 11: The study of the cases n∈ {5,7}; the regions correspond to the non-geodesic optimal
partition and the red arcs are geodesics. Adding midpoints as variables reduces the value from 34.46 to
34.42 for n= 5 and from 68.99 to 68.97 for n= 7.
is not straightforward since we are dealing with a non-differentiable functional. We may expect that
minimizing a p-norm for high pwill get us close to the optimal partition for the maximum. Some
experiments done in this direction indicate that the topology of the optimal partition for the maximum
is the same as the one for the sum, but the boundaries are just slightly moved in order to have the same
eigenvalue for every one of the cells.
n= 3 : three 120◦lens. Confirmation
of the conjecture proposed by Bishop
[6].
optimal cost = 45/4
lens eigenvalue = 15/4.
n= 4 : the optimal partition con-
sists of the tiling generated by a regu-
lar tetrahedron inscribed in the sphere.
optimal cost = 20.635
triangle eigenvalue = 5.1588
n= 5 : two equal equilateral triangles
and three equal rectangles.
optimal cost = 34.46.
triangle eigenvalue = 7.35
rectangle eigenvalue = 6.58.
20
n= 6 : regular tiling generated by the
cube
optimal cost = 48.6.
square eigenvalue = 8.10
n= 7 : two regular pentagons and 5
equal rectangles.
optimal cost = 68.99.
pentagon eigenvalue = 8.61
rectangle eigenvalue = 10.35.
n= 8 : four equal quadrilaterals and
four equal pentagons.
optimal cost = 90.97.
pentagon eigenvalue = 10.80
quadrilateral eigenvalue = 11.94.
n= 9 :3equal squares and 6equal
pentagons.
optimal cost = 115.13.
square eigenvalue = 13.65
pentagon eigenvalue = 12.36.
n= 10 : two equal squares and 8
equal pentagons
optimal cost = 142.26.
square eigenvalue = 15.85
pentagon eigenvalue = 13.94.
n= 11 : one hexagon, two
equal quadrilaterals, eight pentagons
of three types
optimal cost = 175.34.
21
n= 12 : regular tiling generated by
the dodecahedron
optimal cost = 203.71.
pentagon eigenvalue = 16.98
n= 13 : one rectangle, two equal
hexagons, 10 pentagons of three types
optimal cost = 245.37.
n= 14 : two equal hexagons and 12
equal pentagons
optimal cost = 283.59.
hexagon eigenvalue = 17.34
pentagon eigenvalue = 20.74.
n= 15 :3equal hexagons and 12
pentagons of two types
optimal cost = 327.21.
n= 16 :4equal hexagons and 12
equal pentagons
optimal cost = 371.76.
n= 20 :8hexagons and 12 pen-
tagons optimal cost = 585.75.
22
n= 32 :20 equal hexagons and 12
equal pentagons
optimal cost = 1504.16.
Figure 12: Minimal spectral partitions on a torus for n∈[2,9]. The two dimensional pictures represent
flattenings of the torus
9 Conclusions
The method of fundamental solutions seems to be well adapted to the study of the boundary eigenvalue
problems corresponding to the Steklov, Wentzell and Laplace-Beltrami operators. The numerical meth-
Figure 13: Minimal spectral partition on the Banchoff-Chmutov surface of order 4for n∈ {2,4,8}
23
ods presented in this article have good properties regarding their accuracy and speed. This behaviour
encourages their use in the study of shape optimization problems.
The computation of the Steklov or Wentzell spectrum is very fast and precise in the case where the
boundary of the domain is regular enough. It is true that the applicability is restricted to simply connected
domains which do not have cusps. We were able to confirm numerically the conjecture regarding the
optimality of the disk for the first Wentzell eigenvalue proposed in [14].
The method based on fundamental solutions proved useful in the study of the spectral partitions on
manifolds. We are able to obtain results similar to [15] and we also can confirm Bishop’s conjecture
regarding the 3-slice optimal partition of the sphere. Even if numerical evidence suggests that for large n
the optimal partition cells are close to being geodesic polygons, the numerical study of the cases n= 5
and n= 7 allows us to conclude that this fact is not valid in general.
Acknowledgements
The Bishop conjecture was brought to our knowledge by Bernard Helffer. The author wishes to thank
him for the stimulating discussions we had concerning the numerical study of spectral partitions.
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