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arXiv · 2007.05406

Non-existence of axisymmetric optimal domains with smooth boundary for the first curl eigenvalue

Alberto Enciso·Daniel Peralta-Salas
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Abstract

We say that a bounded domain $\Omega$ is optimal for the first positive curl eigenvalue $\mu_1(\Omega)$ if $\mu_1(\Omega)\leq \mu_1(\Omega')$ for any domain $\Omega'$ with the same volume. In spite of the fact that $\mu_1(\Omega)$ is uniformly lower bounded in terms of the volume, in this paper we prove that there are no axisymmetric optimal (and even locally minimizing) domains with $C^{2,\alpha}$ boundary that satisfies a mild technical assumption. As a particular case, this rules out the existence of $C^{2,\alpha}$ optimal axisymmetric domains with a convex section. An analogous result holds in the case of the first negative curl eigenvalue.

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Alberto Enciso, Daniel Peralta-Salas. 2020-07-10. Non-existence of axisymmetric optimal domains with smooth boundary for the first curl eigenvalue. https://arxiv.org/abs/2007.05406

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