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Convex shape optimization for the least biharmonic Steklov eigenvalue

Pedro Ricardo Simão Antunes;  Filippo Gazzola — 2013

ESAIM: Control, Optimisation and Calculus of Variations

The least Steklov eigenvalue for the biharmonic operator in bounded domains gives a bound for the positivity preserving property for the hinged plate problem, appears as a norm of a suitable trace operator, and gives the optimal constant to estimate the -norm of harmonic functions. These applications suggest to address the problem of minimizing in suitable classes of domains. We survey the existing results and conjectures about this topic; in particular,...

Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin laplacian

Pedro Ricardo Simão Antunes;  Pedro Freitas;  James Bernard Kennedy — 2013

ESAIM: Control, Optimisation and Calculus of Variations

We consider the problem of minimising the th-eigenvalue of the Robin Laplacian in R. Although for  = 1,2 and a positive boundary parameter it is known that the minimisers do not depend on , we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on . We derive a Wolf–Keller type result for this problem and show that optimal eigenvalues grow at most with , which is in sharp contrast with the Weyl asymptotics for a fixed domain....

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