Global rigidity of two-dimensional bubbles

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.