A free boundary problem associated with the isoperimetric inequality
  This paper proves a 30 year old conjecture that disks and annuli are the only domains where analytic content - the uniform distance from $\bar{z}$ to analytic functions - achieves its lower bound. This problem is closely related to several well-known free boundary problems, in particular, Serrin's problem about laminar flow of incompressible viscous fluid for multiply-connected domains, and Garabedian's problem on the shape of electrified droplets. Some further ramifications and open questions, including extensions to higher dimensions, are also discussed. 
Ar. Abanov, C. Beneteau, D. Khavinson, and R. Teodorescu
math.CV math-ph math.CA math.MP