The uniqueness of Hill's spherical vortex (Q1087070)

The authors study the free boundary problem \[ r(\frac{1}{r}\psi_ r)_ r+\psi_{zz}= \begin{cases} -\lambda r^ 2f_ 0(\psi) &\text{ in \(A;\)} \\ 0 &\text{ in \(\Pi \setminus A,\)}\end{cases} \] \(\psi |_{r-0}=-k,\quad |_{\partial A}=0\) together with certain asymptotics at infinity. Here \(\Pi =\{(r,z)|\) \(r>0\), \(z\in {\mathbb{R}}\}\), \(f_ 0\geq 0\), and \(\psi\) is a Stokes stream function in cylindrical co-ordinates (no dependence on \(\theta)\). The set \(A\subset \Pi\) is bounded and open, but a priori unknown. A special case of the problem is Hill's problem, in which an explicit solution is known. It is proven that any weak solution to the problem is the explicit solution modulo a translation in z. Such solutions may be obtained as local maximizers of functional.