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

Abstract

We say that a bounded domain $\Omega$ is optimal for the first positive curl eigenvalue ${\mu }_1(\Omega )$ if ${\mu }_1(\Omega )\le {\mu }_1({\Omega }^{\prime })$ for any domain ${\Omega }^{\prime }$ 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.