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Maximization of the Steklov eigenvalues with a diameter constraint. (English) Zbl 1457.49028

Summary: In this paper, we address the problem of maximizing the Steklov eigenvalues with a diameter constraint. We provide an estimate of the Steklov eigenvalues for a convex domain in terms of its diameter and volume, and we show the existence of an optimal convex domain. We establish that balls are never maximizers, even for the first nontrivial eigenvalue that contrasts with the case of volume or perimeter constraints. Under an additional regularity assumption, we are able to prove that the Steklov eigenvalue is multiple for the optimal domain. We illustrate our theoretical results by giving some optimal domains in the plane thanks to a numerical algorithm.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
35P15 Estimates of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

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Ipopt; FreeFem++
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References:

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