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

We say that a bounded domain $Ω$ is optimal for the first positive curl eigenvalue $μ_1(Ω)$ if $μ_1(Ω)\leq μ_1(Ω')$ for any domain $Ω'$ with the same volume. In spite of the fact that $μ_1(Ω)$ 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 $C^{2,α}$ boundary that satisfies a mild technical assumption. As a particular case, this rules out the existence of $C^{2,α}$ optimal axisymmetric domains with a convex section. An analogous result holds in the case of the first negative curl eigenvalue.

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