Vol. 257, No. 1, 2012

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Bounded and unbounded capillary surfaces in a cusp domain

Yasunori Aoki and David Siegel

Vol. 257 (2012), No. 1, 143–165
Abstract

We study asymptotic behavior of the height of a static liquid surface in a cusp domain as modelled by the Laplace–Young capillary surface equation. We introduce a new form of an asymptotic expansion in terms of the functions defining the boundary curves forming a cusp. We are able to address the asymptotic behavior of the capillary surface in cusp domains not previously considered, such as an exponential cusp. In addition, we have shown that the capillary surface in a cusp domain is bounded if the contact angles of the boundary walls forming a cusp are supplementary angles, which implies the continuity of the capillary surface at the cusp.

Keywords
singularity, asymptotic analysis, nonlinear elliptic PDE
Mathematical Subject Classification 2010
Primary: 35A20, 35C20, 35J60, 76B45
Milestones
Received: 17 June 2011
Revised: 16 December 2011
Accepted: 31 December 2011
Published: 19 June 2012
Authors
Yasunori Aoki
Department of Applied Mathematics
University of Waterloo
200 University Avenue West
Waterloo, ON  N2L 3G1
Canada
David Siegel
Department of Applied Mathematics
University of Waterloo
200 University Avenue West
Waterloo, ON  N2L 3G1
Canada