Generalized solutions of the capillary problem
It is known that the capillary problem has not always a solution if the gravity is zero even for bounded domains with smooth boundaries. In the case that a two-dimensional domain has a cusp with an opening angle zero then there is no solution. The author shows that there exists a generalized solution in the sense of M. Miranda. The major difficulty was that previous existence results are based on the assumption of a Lipschitz boundary which is not the case here. The main tool for the proof of the existence is an existence-nonexistence theorem for generalized solutions due to Finn. The results of the author provide hints where the liquid along small touching tubes is located; see a paper of the same author [J. Math. Fluid Mech. 6, 295--310 (2004; Zbl 1050.92041)].
scientific article; zbMATH DE number 2111574