  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.
