On a theorem of Lancaster and Siegel

Danzhu Shi and Robert Finn

Vol. 213 (2004), No. 1, 111–120

DOI: 10.2140/pjm.2004.213.111

Abstract

In establishing conditions for continuity of the height of a capillary surface at a re-entrant corner point of the domain of definition, Lancaster and Siegel introduced a hypothesis of symmetry, which does not appear in corresponding conditions for a protruding corner. We show here that the hypothesis cannot be discarded. Starting with a symmetric configuration for which the surface height is continuous at the corner point in accordance with the hypotheses of those authors, we show that the height can be made discontinuous by an asymmetric domain perturbation that is in an asymptotic sense arbitrarily small, and for which all hypotheses other than that of symmetry remain in force.

Milestones

Received: 8 September 2002

Revised: 14 February 2003

Published: 1 January 2004