Capillary surface discontinuities above reentrant corners

A particular configuration of a vertical capillary tube for which S is the equilibrium interface between two fluids in the presence of a downward pointing gravitational field was investigated. S is the graph a function u whose domain is the (horizontal) cross section gamma of the tube. The mean curvature of S is proportional to its height above a fixed reference plane and lambda is a prescribed constant and may be taken between zero and pi/2. Domains gamma for which us is a bounded function but does not extend continuously to d gamma are sought. Simple domains are found and the behavior of u in those domains is studied. An important comparison principle that has been used in the literature to derive many of the results in capillarity is reviewed. It allows one to deduce the approximate shape of a capillary surface by constructing comparison surfaces with mean curvature and contact angle close to those of the (unknown) solution surface. In the context of nonparametric problems the comparison principle leads to height estimates above and below for the function u. An example from the literature where these height estimates have been used successfully is described. The promised domains for which the bounded u does not extend continuously to the boundary are constructed. The point on the boundary at which u has a jump discontinuity will be the vertext of a re-entrant corner having any interior angle theta pi. Using the comparison principle the behavior of u near this point is studied.