Global rigidity of two-dimensional bubbles

TL;DR

This paper proves the uniqueness of the circular shape for certain two-dimensional vortex problems with surface tension at small Weber numbers, using elliptic PDE analysis and isoperimetric inequalities.

Contribution

It establishes sharp global rigidity results for stationary hollow vortices, supporting a conjecture and classifying solutions based on the Weber number.

Findings

Proves the unit circle is the unique solution for small Weber numbers.

Derives an isoperimetric-isocapacitary inequality related to the problem.

Provides a linear analysis of near-circular solutions.

Abstract

We study stationary hollow vortices with surface tension in two dimensions. Such objects solve an overdetermined elliptic free boundary problem in an exterior domain, with an additional boundary condition involving mean curvature and the Neumann trace. We prove sharp global rigidity of the unit circle for small Weber numbers, supporting a conjecture of Crowdy and Wegmann. This elliptic problem describes critical points of the sum of the perimeter and the logarithmic potential energy of bounded sets. We prove an isoperimetric-isocapacitary inequality and classify, in terms of the Weber number, when the unit disk is the unique solution to the associated convexity-constrained variational problem. Furthermore, a linear analysis gives precise description into close-to-circular solutions for both problems.