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

TL;DR

This paper proves that no smooth, axisymmetric domain with certain regularity can minimize the first curl eigenvalue among domains of the same volume, ruling out the existence of optimal shapes with convex sections.

Contribution

It establishes the non-existence of smooth, axisymmetric optimal domains for the first curl eigenvalue under mild regularity assumptions.

Findings

No axisymmetric $C^{2,eta}$ domains are optimal for the first curl eigenvalue.

Optimal domains with convex sections do not exist under the given regularity.

Analogous non-existence results hold for the negative curl eigenvalue.

Abstract

We say that a bounded domain $\Omega$ is optimal for the first positive curl eigenvalue $\mu_1(\Omega)$ if $\mu_1(\Omega)\leq \mu_1(\Omega')$ 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.