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.

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 Gauss's circle problem.

CONTENTS 1. Introduction 2. The Numerical Method 3. Results for λ 1 on Polygons 4. One-Term Bounds 5. Two-Term Bounds 6. Discussion Acknowledgments References

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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 2 and 3 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).

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.

We consider the problem of minimising the n th −eigenvalue of the Robin Laplacian in R N . Although for n = 1, 2 and a positive boundary parameter α it is known that the minimisers do not depend on α, we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on α. We derive a Wolf-Keller type result for this problem and show that optimal eigenvalues grow at most with n 1/N , which is in sharp contrast with the Weyl asymptotics for a fixed domain. We further show that the gap between consecutive eigenvalues does go to zero as n goes to infinity. Numerical results then support the conjecture that for each n there exists a positive value of αn such that the n th eigenvalue is minimised by n disks for all 0 < α < αn and, combined with analytic estimates, that this value is expected to grow with n 1/N .

SUMMARYIn this paper we study the application of the method of fundamental solutions (MFS) to the numerical calculation of the eigenvalues and eigenfunctions for the 2D bilaplacian in simply connected plates. 251-266), which leads to very good numerical results for simply connected domains. A main part of this paper is devoted to the numerical analysis of the method, presenting a density result that justifies the application of the MFS to the eigenvalue biharmonic equation for clamped plate problems. We also present a bound for the eigenvalues approximation error, which leads to an a posteriori convergence estimate.