Some Unusual Comparison Properties of Capillary Surfaces

Under typical physical conditions, the solution of the capillarity equation for a tube of circular section D will always exceed over D the solution obtained for a concentric tube of the same material and larger radius. We address here a question raised by M. Miranda, as to whether a solution over a general domain D0 will exceed, over that section, the solution over any domain D1 strictly containing D0. We show that whenever a domain D1 admits a zero gravity solution surface in a variational sense for the given contact angle, and has at some point a boundary curvature inward directed and exceeding the ratio of perimeter to area of the section, there is then a subdomain D0 for which a negative answer appears for all sufficiently small gravity g; that occurs with height differences inversely proportional to g, uniformly over D0. Under other conditions, positive answers appear. We provide an example in which the limiting behavior as g → 0 reverses in a discontinuous way, with smooth infinitesimal change of ∂D0. Remarkably, the discontinuous change occurs at a circular cylinder configuration, for which one normally expects stable behavior. The discussion includes some results that seem to have general geometric interest; notably, we characterize in Theorem 5 all convex domains containing a disk, and for which the ratio of perimeter to area is not less than for the disk.