Optimal convex domains for the first curl eigenvalue

We prove that there exists a bounded convex domain $\Omega \subset \mathbf{R}^3$ of fixed volume that minimizes the first positive curl eigenvalue among all other bounded convex domains of the same volume. We show that this optimal domain cannot be analytic, and that it cannot be stably convex if it is sufficiently smooth (e.g., of class $C^{1,1}$). Existence results for uniformly H\"older optimal domains in a box (that is, contained in a fixed bounded domain $D \subset \mathbf{R}^3$) are also presented.