Consider a cylinder of
homogenous material closed at one end by a base of general cross section Ω and
partly filled with liquid. We want to find conditions under which in the absence of
gravity the liquid can cover Ω and is in mechanical equilibrium.
If the liquid can cover Ω, then the liquid surface is a graph over the base. In
general, the surface has constant mean curvature and makes constant angle with
the bounding wall. Even if Ω is convex analytic, such a surface may not
exist. However, it is the case when Ω is piecewise smooth that interests us.
In this case, the interior angles at the corners play an important role. It
turns out that the existence of the liquid surface as a graph over the base
can be characterized by the nonexistence of a certain subsidiary variational
problem.
