Capillary surfaces at a reentrant corner (Q883844)

The paper concerns capillary surfaces over domains with corners when the opening angle exceeds \(\pi\) (reentrant corner). It is known that all solutions are locally bounded near the corner, but very different kinds of behaviour must be expected, in particular a discontinuous behaviour at the corner. The author characterizes all possible modes of behaviour subjected to a conjecture of Concus and Finn for the protruding angle case. This conjecture says that capillary surface over a domain with a corner with opening angle \(2\alpha\), \(0<2\alpha<\pi\), is discontinuous near the corner if the contact angles on the adjacent curves of the corner differ by more then \(\pi-2\alpha\). The interesting results of the paper are based, in particular, on a detailed study of associated generalized solutions in the sense of M.~Miranda and on the work of E.~Giusti concerning minimal graphs. The results of the paper support the conjecture of Concus and Finn since the results based on this conjecture agree with known results for reentrant corners.