Pedro R. S. Antunes

Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians

We perform a numerical optimization of the first ten nontrivial eigenvalues of the Neumann Laplacian for planar Euclidean domains. The optimization procedure is done via a gradient method, while the computation of the eigenvalues themselves is done by means of an efficient meshless numerical method which allows for the computation of the eigenvalues for large numbers of domains within a reasonable time frame. The Dirichlet problem, previously studied by Oudet using a different numerical method, is also studied and we obtain similar (but improved) results for a larger number of eigenvalues. These results reveal an underlying structure to the optimizers regarding symmetry and connectedness, for instance, but also show that there are exceptions to these preventing general results from holding.

New Bounds for the Principal Dirichlet Eigenvalue of Planar Regions

We present a numerical study for the first Dirichlet eigenvalue of certain classes of planar regions. Based on this, we propose new types of bounds and establish several conjectures regarding the dependence of this eigenvalue on the perimeter and the area.

A numerical study of the spectral gap

We present a numerical study of the spectral gap of the Dirichlet Laplacian, γ(K) = λ2(K) − λ1(K), of a planar convex region K. Besides providing supporting numerical evidence for the long-standing gap conjecture that γ(K) ≥ 3π2/d2(K), where d(K) denotes the diameter of K, our study suggests new types of bounds and several conjectures regarding the dependence of the gap not only on the diameter, but also on the perimeter and the area. One of these conjectures is a stronger version of the gap conjecture mentioned above. A similar study is carried out for the quotient of the first two Dirichlet eigenvalues.

Optimal spectral rectangles and lattice ellipses

We consider the problem of minimizing the kth eigenvalue of rectangles with unit area and Dirichlet boundary conditions. This problem corresponds to finding the ellipse centred at the origin with axes on the horizontal and vertical axes with the smallest area containing k integer lattice points in the first quadrant. We show that, as k goes to infinity, the optimal rectangle approaches the square and, correspondingly, the optimal ellipse approaches the circle. We also provide a computational method for determining optimal rectangles for any k and relate the rate of convergence to the square with the conjectured error term for Gausss circle problem.

A meshfree numerical method for acoustic wave propagation problems in planar domains with corners and cracks

The numerical solution of acoustic wave propagation problems in planar domains with corners and cracks is considered. Since the exact solution of such problems is singular in the neighborhood of the geometric singularities the standard meshfree methods, based on global interpolation by analytic functions, show low accuracy. In order to circumvent this issue, a meshfree modification of the method of fundamental solutions is developed, where the approximation basis is enriched by an extra span of corner adapted non-smooth shape functions. The high accuracy of the new method is illustrated by solving several boundary value problems for the Helmholtz equation, modelling physical phenomena from the fields of room acoustics and acoustic resonance.

THE METHOD OF FUNDAMENTAL SOLUTIONS APPLIED TO SOME INVERSE EIGENPROBLEMS

In this work we address the application of the method of fundamental solution (MFS) as a forward solver in some shape optimization problems in two- and three-dimensional domains. It is well known (Kacs problem) that a set of eigenvalues does not determine uniquely the shape of the domain. Moreover, even the existence problem is not well defined due to the Ashbaugh--Benguria inequality. Although these results constitute counterexamples in the general problem of shape determination from the eigenfrequencies, we can address simpler questions in shape determination using the MFS. For instance, we apply the MFS to build domains that include a specific finite set of eigenvalues, or that have an eigenmode that verifies some prescribed conditions---as a particular case, an eigenmode that defines a certain nodal line.

On the range of the first two Dirichlet and Neumann eigenvalues of the Laplacian

In this paper, we study the set of points in the plane defined by {(x, y)=(λ1(Ω), λ2(Ω)), |Ω|=1}, where (λ1(Ω), λ2(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.

On the inverse spectral problem for Euclidean triangles

We consider the inverse spectral problem for the Laplace operator on triangles with Dirichlet boundary conditions, providing numerical evidence to the effect that the eigenvalue triplet (λ1,λ2,λ3) is sufficient to determine a triangle uniquely. On the other hand, we show that other combinations such as (λ1,λ2,λ4) will not be enough, and that there will exist at least two triangles with the same values on these triplets.

Reducing the ill conditioning in the method of fundamental solutions

The method of fundamental solutions (MFS) is a meshless method for solving boundary value problems with some partial differential equations. It allows to obtain highly accurate approximations for the solutions assuming that they are smooth enough, even with small matrices. As a counterpart, the (dense) matrices involved are often ill-conditioned which is related to the well known uncertainty principle stating that it is impossible to have high accuracy and good conditioning at the same time. In this work, we propose a technique to reduce the ill conditioning in the MFS, assuming that the source points are placed on a circumference of radius R. The idea is to apply a suitable change of basis that provides new basis functions that span the same space as the MFS’s, but are much better conditioned. In the particular case of circular domains, the algorithm allows to obtain errors close to machine precision, with condition numbers of order O(1), independently of the number of points sources and R.

Bounds and extremal domains for Robin eigenvalues with negative boundary parameter

Abstract We present some new bounds for the first Robin eigenvalue with a negative boundary parameter. These include the constant volume problem, where the bounds are based on the shrinking coordinate method, and a proof that in the fixed perimeter case the disk maximises the first eigenvalue for all values of the parameter. This is in contrast with what happens in the constant area problem, where the disk is the maximiser only for small values of the boundary parameter. We also present sharp upper and lower bounds for the first eigenvalue of the ball and spherical shells. These results are complemented by the numerical optimisation of the first four and two eigenvalues in two and three dimensions, respectively, and an evaluation of the quality of the upper bounds obtained. We also study the bifurcations from the ball as the boundary parameter becomes large (negative).