Uniqueness for the Circular Vortex-Pair in a Uniform Flow

In the x-y-plane, there is a well-known example of an exact formula for a steady ideal fluid flow, that is symmetric in the x-axis, that contains a pair of vortices bounded by a single circle, and whose far-field approaches a uniform flow in the x-direction. The stream function in this case has the form psi - y where psi obeys the partial differential equation -Δ psi = (psi - y)₊ for y > 0 while approaching 0 at infinity and vanishing on the x-axis. The present work proves a uniqueness theorem for this boundary-value problem, by using the consequences of the maximum principle to establish symmetry properties of its solutions. An isoperimetric property of the flow is also established.