Global Rigidity of Two-Dimensional Bubbles

We study bubbles, which are 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 curvature and the Neumann trace. Motivated by a conjecture of Crowdy and Wegmann, we prove that every solution prescribed by a Jordan curve is circular for $0\le \text{We}\le 3$ , where 3 is the first bifurcation value. This range is sharp: the bifurcation branch at $\text{We}=3$ contains non-circular solutions with $\text{We}>3$ arbitrarily close to 3. This elliptic problem describes critical points of the sum of the perimeter and the logarithmic potential energy of compact sets. We prove an area-constrained perimeter-capacity inequality and show that, in the associated convexity-constrained variational problem, the unit disc is the unique minimiser up to translation if and only if $0\le \text{We}\le 3$ .