Optimal convex domains for the first curl eigenvalue

TL;DR

This paper establishes the existence of convex domains in three-dimensional space that minimize the first curl eigenvalue for a fixed volume, revealing properties about their smoothness and analyticity.

Contribution

It proves the existence of volume-fixed convex domains minimizing the first curl eigenvalue and analyzes their regularity and analyticity properties.

Findings

Optimal convex domains exist for the first curl eigenvalue at fixed volume.

Such optimal domains cannot be analytic.

Optimal domains cannot be stably convex if sufficiently smooth.

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