Optimal convex domains for the first curl eigenvalue

Abstract

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 Hölder optimal domains in a box (that is, contained in a fixed bounded domain $D\subset {\mathbf{R}}^3$) are also presented.

References in corpus (1)

• Existence and characterisation of magnetic energy minimisers on oriented, compact Riemannian 3-manifolds with boundary in arbitrary helicity classes

Cited by in corpus (2)

• Existence of optimal domains for the helicity maximisation problem among domains satisfying a uniform ball condition
• Asymptotic windings, surface helicity and their applications in plasma physics