Singular solutions of the capillary problem
Robert Finn and Robert Weston Neel
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Submission date: 16. Nov. 1998
published in: Journal für die reine und angewandte Mathematik, 512 (1999), p. 1-25
DOI number (of the published article): 10.1515/crll.1999.053, 01/07/1999
with the following different title: C-singular solutions of the capillary problem
MSC-Numbers: 76B45, 53A10, 53C80, 49Q10, 35J60, 35B65
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The problem of determining the surface interface of fluid partly filling a semi-infinite capillary tube closed at one end is considered, in the absence of gravity. It is known that solutions of the formal equations for a solution as a graph covering the base do not in general exist. In this paper it is shown that whenever smooth solutions fail to exist there will nevertheless exist a solution in a singular sense suggested by physical intuition. In some configurations of particular interest, the procedure leads to unique determination of the singular solution. However, uniqueness cannot in general be expected, as is shown by example. Further examples show a) that singular solutions may appear also when smooth solutions exist, and b) they may fail to occur in that case, depending on the particular geometry.