Numerical results for extremal problem for eigenvalues of the Laplacian

Some of these shape optimizations have already been solved. The rst Dirichlet eigenvalue is minimized by the ball, as proven by Faber and Krahn [262, 428]. The second Dirichlet eigenvalue is minimized by two balls of the same volume [429]. In two dimensional case, it has long been conjectured that the ball minimizes λ3(Ω), but there has not been much progress in this direction. For higher eigenvalues, not much is known and even the existence of minimizers among quasi-open sets has only been proven quite recently (see Chapter 2) and [146, 496]. It is worth mentioning the work by Berger [88] who proved that for i > 4, the i-th Dirichlet eigenvalue is not minimized by any union of balls. For the Neumann problem, we have μ1(Ω) = 0. The second eigenvalue, μ2(Ω), is maximized by the ball. The result had been conjectured by Kornhauser and Stakgold in [424] and was proved by Szegö in [592] for Lipschitz simply connected planar domains and generalized byWeinberger in [614] to arbitrary domains, and any dimension. More recently, Girouard, Nadirashvili and Polterovich proved that the maximum of μ3(Ω) among simply connected bounded planar domains is attained by two disjoint balls of equal area [299]. Recently, many works have addressed numerical approaches that propose candidates for the optimizers for these and related spectral problems, and to suggest conjectures about their qualitative properties [28, 103, 516, 517, 518, 519]. In the next two sections we describe brie y two of these approaches which have been successful for spectral problems. We rst introduce some global optimization tools to provide a good initial guess of the optimal pro le. This step does not require any topological information on the set but is restricted to a small class of shapes.Moreover, since this approach relies only on the parametrization of the space of shapes, any global algorithm can be used as a black box solver to nd a starting candidate for the


Numerical results for extremal problem for eigenvalues of the Laplacian
We consider in this chapter shape optimization problems for Dirichlet and Neumann eigenvalues, and Some of these shape optimizations have already been solved.The rst Dirichlet eigenvalue is minimized by the ball, as proven by Faber and Krahn [262,428].The second Dirichlet eigenvalue is minimized by two balls of the same volume [429].In two dimensional case, it has long been conjectured that the ball minimizes λ (Ω), but there has not been much progress in this direction.For higher eigenvalues, not much is known and even the existence of minimizers among quasi-open sets has only been proven quite recently (see Chapter 2) and [146,496].It is worth mentioning the work by Berger [88] who proved that for i > , the i-th Dirichlet eigenvalue is not minimized by any union of balls.
For the Neumann problem, we have µ (Ω) = .The second eigenvalue, µ (Ω), is maximized by the ball.The result had been conjectured by Kornhauser and Stakgold in [424] and was proved by Szegö in [592] for Lipschitz simply connected planar domains and generalized by Weinberger in [614] to arbitrary domains, and any dimension.More recently, Girouard, Nadirashvili and Polterovich proved that the maximum of µ (Ω) among simply connected bounded planar domains is attained by two disjoint balls of equal area [299].
Recently, many works have addressed numerical approaches that propose candidates for the optimizers for these and related spectral problems, and to suggest conjectures about their qualitative properties [28,103,516,517,518,519].
In the next two sections we describe brie y two of these approaches which have been successful for spectral problems.We rst introduce some global optimization tools to provide a good initial guess of the optimal pro le.This step does not require any topological information on the set but is restricted to a small class of shapes.Moreover, since this approach relies only on the parametrization of the space of shapes, any global algorithm can be used as a black box solver to nd a starting candidate for the Pedro R. S. Antunes: Grupo de Física Matemática da Universidade de Lisboa, Portugal, E-mail: prantunes@fc.ul.ptEdouard Oudet: Laboratoire Jean Kuntzmann (LJK), Université Joseph Fourier et CNRS, Tour IRMA, BP 53, 51 rue des Mathématiques, 38041 Grenoble Cedex 9 -France, E-mail: edouard.oudet@imag.frlocal procedure.We then describe the method of fundamental solutions which is able in a second stage to both identify and evaluate precisely shapes which are locally optimal.

Some tools for global numerical optimization in spectral theory
Global optimality is perhaps one of the most challenging aspects in numerical shape optimization.Many approaches have been developed to tackle this di culty: stochastic algorithms, multi scale methods, relaxation, homogenization, etc.In this section we rst recall brie y an historical method developed to apply a genetic algorithm in the framework of shape optimization.It has the bene t of dramatically reducing the number of degrees of freedom, which makes the global optimization more e cient, it has the drawback of parametrizing non smooth pro les.
In the following we introduce a very naive approach based on implicit representation which makes it possible both to reduce the number of unknowns and to generate smooth shapes.In a second step we describe a new simple idea to restrict the search space in spectral optimization to one where homogeneous functionals are frequently involved.

. An historical approach: Genetic algorithm and Voronoi cells
Consider a given grid covering a search domain in which we look for an optimal shape Ω.A purely discrete numerical approach consists of associating to every node of the grid a boolean value which expresses the fact that the node is or is not in the set Ω.
In the 90's, Allaire et al.
[14] introduced a discretization framework in shape optimization based on Voronoi cells associated to a set of points independent of the grid.In this setting a grid point of the search space is considered to be a part of Ω if the seed of its Voronoi cell has a True boolean value.The interesting part of this method is that the complexity of the approach is not anymore related to the grid size but only on the number of seed points.This crucial distinction makes it possible to compute state solutions of partial di erential equations with a reasonable precision whereas the number of unknowns is not too large.A drawback of the method is the non smoothness of Voronoi cells.Due to its polygonal faces, an large number of Voronoi seeds may be required to approximate smooth shapes.In the speci c context of eigenvalues where some smoothness is expected, this kind of discretization is not optimal.

. Smooth pro les with few parameters
In this section related to global spectral optimization we consider the minimization of functional of the type where F is some smooth xed nite dimensional function.We address the optimization problem min Depending on the context we may consider additional geometrical constraints imposed on the set Ω like a xed measure or boundary measure, connectivity, convexity, etc.
Identifying the global optimal solution of a non convex and sometimes non smooth cost function is in almost all cases an untenable task.In order to decrease the complexity of this problem we introduce a reduction of the number mp of parameters which still allows a precise computation of the cost function.Since we use black box stochastic algorithm we need to reduce the number of parameters to mp ≤ for instance.More precisely we develop this dimension reduction by introducing the two following steps.
In the spirit of level set methods, we rst parameterize the set of shape as the level sets of truncated Fourier series.Contrary to standard boundary parametrization where the number of degrees of freedom is related to the number of boundary points in the mesh, our parametrization has the complexity of the number of terms in the Fourier series.Moreover, the very basic and crucial observation is the fact that this complexity is not related to the precision of the eigenvalue approximation.
Our second improvement is related to the reduction of the size of the search space.We reduce the complexity of the optimization process by substituting the cost evaluation of a given shape by the optimal value associated to the best homothetic connected components.Essentially it relies on the two following classical properties: Proposition 1.1 (Homogeneity).Let α > be a real.Then for all integers j, (5) Notice that homogeneity can be used to a transform constrained problem like The second property makes it possible to compute more e ciently optimal pro les with multiply connected components: Proposition 1.2 (Multiply connected components).Let Ω , . . ., Ωm be the connected components of Ω.For i = , . . ., m, let Λ i be the set of the eigenvalues of −∆ on Ω i for Dirichlet or Neumann condition.Then the set of the eigenvalues of ∆ is Λ = m i= Λ i .
We now describe our discretization of the search space for d = .The generalization to higher dimensions is straightforward.Let us consider coe cients a i,j such that {a i,j } = n (the number of coe cients).Then, de ne the Fourier series a i,j sin(πix ) sin(πjx ) + , where x = (x , x ) ∈ [ , ] .( 8) Notice that we add the constant value 1 to previous sum so that the function is non-zero on ∂Ω.That is the level set domain does not intersect the boundary.We now de ne F : R n → P(R ) by Finally we build the sets Notice that the topology or more speci cally the number of connected components of Ω {a i,j } is not imposed by the algorithm.
In practice, B = [ , ] is meshed by a Cartesian grid.Φ {a i,j } is evaluated at every point of the mesh and a linear interpolation is carried out to approximate F({a i,j }).Through this discretization we associate a polygon Ω pol {a i,j } to every Ω {a i,j } .We then de ne the cost function associated to the parameters a i,j by Where the symbol expresses the fact we do not optimize the true eigenvalues of the polygons int this process but rather the nite element approximation of these eigenvalues.Finally, it is standard to approximate the cost function G(Ω pol {a i,j } ) by classical Finite Element Methods.Notice that every evaluation requires us to construct a new mesh adapted to the polygon Ω pol {a i,j } since linear interpolation generates meshes of very bad quality.In all our experiments we xed approximately the number of simplices per evaluation to obtain comparable results.

. A fundamental complexity reduction: optimal connected components
We detailed in previous section a way to parametrize multi-connected shapes with few parameters.As it has been explained, every cost evaluation requires to mesh the new domain and to solve the associated discrete spectral optimization problem.This step can be very time-consuming especially in the case of 3 dimension computations.
To tackle this di culty, we would like to use the homogeneity of eigenmodes to investigate homothetical components in one single cost evaluation.Actually, due to the homogeneity of eigenmodes the computation of these modes associated to one geometrical con guration can be used to deduce the cost function of any domain made of homothetic components.
From properties 1.1 and 1.2 we obtain that if Ω = α Ω ∪ α Ω (disjoint union) then The crucial fact observation is that the computation of λ j (α Ω ∪ α Ω ) and µ j (α Ω ∪ α Ω ) is equivalent to the sorting operation of the union of the two sets Let Ω be a xed set with m connected components.Let us associate to Ω a new cost which is the best value obtained with respect to its homothetical connected components.More precisely we de ne: where Notice that due to the translation invariance of the problem we can always assume that every connected components are disjoint.As a consequence, we can associate to a xed geometrical con guration with m connected components a new cost de ned by ( 11).This small scale global problem (11) can be solved very e ciently by using global algorithm like Lipschitz optimization (see for instance [406]).Moreover, since the number of unknowns is small (the number of expected connected components) and the cost evaluation is pretty fast, this global optimization problem can be solved very quickly with respect to a nite element evaluation.

Numerical approach using the method of fundamental solutions
The eigenvalue problem for the Laplace operator is equivalent to obtaining the reso- (13) Among other numerical approaches, these eigenvalue problems can be solved by the Method of Fundamental Solutions (MFS) [20,408].The MFS is a Tre tz type method, where the particular solutions are point sources centered outside the domain.More precisely, denoting by .the Euclidean norm in R d , we take a fundamental solution of the Helmholtz equation for the two-dimensional case, where H ( ) is the rst Hankel function and in the three-dimensional case.We have (∆ + κ )Φκ = −δ, where δ is the Dirac delta distribution.For a given frequency κ, we consider a basis built with point sources where y j ∈ Ω.By Γ = ∂ Ω, we will denote an admissible source set, for instance, the boundary of a bounded open set Ω ⊃ Ω, with Γ surrounding ∂Ω.
The MFS approximation is a linear combination where the source points y j are placed on an admissible source set.The approximation of an eigenfunction by a MFS linear combination can be justi ed by density results (e.g.[20]), Theorem 2.1.Consider Γ = ∂ Ω, an admissible source set.Then, Next we give a brief description of the application of the MFS for determining the eigensolutions for a given shape Ω.For details, see [19] and [25] respectively for the two and three-dimensional cases.The eigensolutions are obtained in two steps.First, we calculate an approximate eigenfrequency κ and then, for that frequency, we obtain the approximation for the eigenfunction.
We de ne m collocation points x i almost uniformly distributed on the boundary ∂Ω and for each of those points we de ne a corresponding source point, where α is a positive parameter and n i is the unitary outward normal vector at the point x i .Imposing the Dirichlet boundary conditions at the boundary points we obtain the system m j= α j Φκ(x i − y j ) = . (18) A straightforward procedure for calculating the eigenfrequencies is to nd the values κ for which the m × m matrix is singular.The Neumann case is similar.
To obtain an eigenfunction associated with a certain resonant frequency κ we use a collocation method on n + points, with x , • • • , xn on ∂Ω and a point x n+ ∈ Ω.The eigenfunction is approximated by an MFS approximation, and to exclude the trivial solution ũ(x) ≡ , the coe cients α j are determined by solving the system ũ(x i ) = δ i,n+ , i = , . . ., n + where δ i,j is the Kronecker delta.An advantage of the Method of Fundamental Solutions approach with respect to Finite Element Method approach is the fact that we can calculate rigorous bounds for the errors associated to approximate eigenvalues.In the Dirichlet case, the error can be estimated by using an a posteriori bound due to Fox, Henrici and Moler (cf.[272]).This result provides upper bounds for the errors of the approximations obtained in several methods of particular solutions and was also used in rigorous proofs (eg. [19, 93, 461]).Next, we de ne the class of admissible domains for the shape optimization.Note that if for some i = , , ..., the optimizer of the problems (1) or ( 2) is disconnected, by ), each of the connected components are optimizers of a lower eigenvalue.Thus, we will focus on the numerical solution of the shape optimization problem among connected domains and then compare this optimal value against the optimal value obtained for disconnected sets by using Wolf-Keller theorem.
We consider the functions For the Neumann case, assuming that Ω is of class C , µ k is simple and u is the associated normalized eigenfunction, we have The menagerie of the spectrum In this section we present the main numerical results that we gathered for the solution of the shape optimization problems (1) and (2).In Figure 1 we plot the minimizers of the rst 15 Dirichlet eigenvalues.
Table ?? shows the optimal Dirichlet eigenvalues, together with the corresponding multiplicity of each optimal eigenvalue.In Figure 2 we plot the maximizers of the rst 10 (non trivial) Neumann eigenvalues in the class of unions of simply connected domains.
Table ?? shows the optimal Neumann eigenvalues and the corresponding optimal multiplicity.
Next, we present some numerical results for the shape optimization problems (1) and ( 2) with three-dimensional domains.In Figure 3 we plot the 3D minimizers of the rst 10 Dirichlet eigenvalues.Table ?? shows the optimal 3D Dirichlet eigenvalues and the corresponding multiplicity.Figure 4 and Table ?? show similar results for Neumann eigenvalues.

Open problems
The numerical results that we obtained suggest some conjectures to non trivial solutions of the Helmholtz equation up ≡ : ∆up + κ p up = in Ω, up = or ∂n u = on Γ.

problem 4. 1 .
Prove that the d-dimensional ball minimizes λ d+ among all ddimensional sets of a xed volume.Open problem 4.2.Prove that the d-dimensional minimizer of λ d+ among all ddimensional sets of a xed volume is the union of two balls whose radii are in the ratioj d , j d − , , where j n,k is the k-th zero of the Bessel function Jn.

Table 4 .
The optimal 3D Neumann eigenvalues and the corresponding multiplicity.