On existence criteria for capillary free surfaces without gravity (Q1083687)

Consider a cylinder of homogeneous material closed at one end by a base of general cross section \(\Omega\) and partly filled with liquid. We want to find conditions under which in the absence of gravity the liquid can cover \(\Omega\) and is in mechanical equilibrium. If the liquid can cover \(\Omega\), 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 \(\Omega\) is convex analytic, such a surface maynot exist. However, it is the case when \(\Omega\) 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 be characterized by the nonexistence of a certain subsidiary variational problem.