Numerical minimization of eigenmodes of a membrane with respect to the domain

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Eigenvalues of poly-harmonic operators on variable domains (2012, background): “Moreover, by looking at the interesting numerical results presented in Oudet [23], one may get the idea that balls are not much relevant in the analysis of higher eigenvalues.”
The Method of Fundamental Solutions Applied to the Calculation of Eigenfrequencies and Eigenmodes of 2D Simply Connected Shapes (2005, methodology): “In Oudet (2004) it was proposed an optimization algorithm which allowed to obtain a domain with a second eigenvalue which is smaller than the value of the stadium.”
Lipschitz Regularity of the Eigenfunctions on Optimal Domains (2013, result): “It was recently proved by Berger and Oudet2 that the later result holds for all k ∈ N large enough, which confirmed the previous numerical results obtained in [25] and [3].”