Optimisation of Eigenvalues of the Dirichlet Laplacian with a Surface Area Restriction

We perform a numerical optimisation of the low frequencies of the Dirichlet Laplacian with perimeter and surface area restrictions, in two and 3-dimensions, respectively. In the former case, we handle the first 50 eigenvalues and measure the rate at which the corresponding optimisers approach the disk, while in the latter we optimise the first twenty eigenvalues. We derive theoretical compatibility conditions which must be satisfied by a sequence of optimisers and test our numerical results against these. We also consider the cases of rectangles with a fixed perimeter and parallelepipeds with a surface restriction for which we compute the first $$10^7$$107 and $$10^6$$106 optimal eigenvalues, respectively. In this context, we prove convergence to the cube in any dimensions and compare the numerical results with our theoretical estimates for the rate of convergence.

[1]  Carlos J. S. Alves,et al.  The Method of Fundamental Solutions Applied to Some Inverse Eigenproblems , 2013, SIAM J. Sci. Comput..

[2]  Pedro Freitas,et al.  Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians , 2012, J. Optim. Theory Appl..

[3]  Edouard Oudet,et al.  Numerical minimization of eigenmodes of a membrane with respect to the domain , 2004 .

[4]  Bozhidar Velichkov,et al.  Existence and Regularity of Minimizers for Some Spectral Functionals with Perimeter Constraint , 2013, 1303.0968.

[5]  C. Kao,et al.  Minimal Convex Combinations of Three Sequential Laplace-Dirichlet Eigenvalues , 2013 .

[6]  Pedro R. S. Antunes.  Numerical calculation of eigensolutions of 3D shapes using the method of fundamental solutions , 2011 .

[7]  Lower bounds for the eigenvalues of the fixed vibrating membrane problems , 1984 .

[8]  A. Berger.  Optimisation du spectre du Laplacien avec conditions de Dirichlet et Neumann dans R² et R³ , 2015 .

[9]  B. Colbois,et al.  Extremal eigenvalues of the Laplacian on Euclidean domains and closed surfaces , 2014, 1403.1993.

[10]  Range of the first three eigenvalues of the planar Dirichlet Laplacian , 2002, math/0203231.

[11]  Yu Safarov,et al.  The Asymptotic Distribution of Eigenvalues of Partial Differential Operators , 1996 .

[12]  George Polya,et al.  On the Eigenvalues of Vibrating Membranes(In Memoriam Hermann Weyl) , 1961 .

[13]  Édouard Oudet,et al.  Qualitative and Numerical Analysis of a Spectral Problem with Perimeter Constraint , 2016, SIAM J. Control. Optim..

[14]  P. Freitas,et al.  Asymptotic behaviour of optimal spectral planar domains with fixed perimeter , 2013 .

[15]  G. Buttazzo,et al.  Minimization of $\lambda_2(\Omega)$ with a perimeter constraint , 2009, 0904.2193.

[16]  L. Hörmander,et al.  THE ASYMPTOTIC DISTRIBUTION OF EIGENVALUES OF PARTIAL DIFFERENTIAL OPERATORS (Translations of Mathematical Monographs 155) , 1998 .

[17]  Pedro R. S. Antunes,et al.  Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian , 2012, 1204.0648.

[18]  Pedro R. S. Antunes,et al.  Optimal spectral rectangles and lattice ellipses , 2013, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.