Global rigidity of two-dimensional bubbles
Lukas Niebel
TL;DR
The paper analyzes a two-dimensional free-boundary problem for stationary hollow vortices with surface tension, governed by the Weber number $\mathrm{We}$. By combining Pohozaev-type identities, Minkowski formulae, and isoperimetric inequalities, it proves global rigidity of the unit disk for $0\le\mathrm{We}\le 2$, showing circles are the unique solutions in this regime; a linear Fourier analysis via the exterior Dirichlet-to-Neumann map yields precise local rigidity results, excluding $\mathrm{We}\in\{3,4,5,\dots\}$. It also connects the overdetermined problem to a variational energy $\mathcal{F}_{\mathrm{We}}(E)=\mathrm{We}\,\pi\mathcal{I}(E)+\mathcal{P}(E)$, establishing global minimality of the disk for $We\le2$ (under Jordan-curve constraint) and demonstrating non-minimality of the circle for $We>3$ through a second-order instability along elliptical perturbations. Together, these results support the Crowdy–Wegmann conjecture on circle rigidity for small We and illuminate the interplay between free-boundary PDEs and logarithmic-energy variational problems.
Abstract
We study stationary hollow vortices with surface tension in two dimensions. Such objects are solutions to an overdetermined elliptic free boundary value problem in an exterior domain, where an additional condition involving the mean curvature and the Neumann trace on the boundary is imposed. We prove global rigidity of the circle for small Weber numbers, supporting a conjecture of Crowdy and Wegmann. This elliptic problem describes critical points of the sum of perimeter and the logarithmic potential energy of bounded sets. The variational problem is ill-posed in general, but we recover the global rigidity for small Weber numbers in the class of sets bounded by a Jordan curve. A linear analysis gives precise insights into close-to-circular solutions for both problems.