  Inspired by a planar partitioning problem involving multiple improper
chambers, this article investigates using classical techniques what can be said
of the existence, uniqueness, and regularity of minimizers in a certain
free-endpoint isoperimetric problem. By restricting to curves which are
expressible as graphs of functions, a full existence-uniqueness-regularity
result is proved using a convexity technique inspired by work of Talenti. The
problem studied here can be interpreted physically as the identification of the
equilibrium shape of a sessile liquid drop in half-space (in the absence of
gravity). This is a well-studied variational problem whose full resolution
requires the use of geometric measure theory, in particular the theory of sets
of finite perimeter, but here we present a more direct, classical geometrical
approach. Conjectures on improper planar partitioning problems are presented
throughout.
