Existence and behavior of the radial limits of a bounded capillary surface at a corner

Kirk Lancaster and David Siegel

Vol. 176 (1996), No. 1, 165–194

DOI: 10.2140/pjm.1996.176.165

Abstract

Consider a bounded capillary surface defined on a two-dimensional region Ω that has a corner point at O, with opening angle 2α. If the contact angle is bounded away from 0 and π, then the radial limits exist as O is approached from any direction in Ω. If the contact angle approaches limiting values as O is approached along each portion of the boundary, then there exist “fans” of directions adjacent to the two tangent directions at O in which the radial limits are constant. Other properties of the radial limit function are given and these results are used to show continuity of the solution up to O under certain conditions. For a convex corner, the solution is continuous up to O when the limiting angles γ₀⁺, γ₀⁻ satisfy |π − γ₀⁺ − γ₀⁻| < 2α and 2α + |γ₀⁺ − γ₀⁻|≤ π.

Mathematical Subject Classification 2000

Primary: 58E12

Secondary: 53A10, 35J60

Milestones

Received: 9 June 1994

Revised: 10 May 1995

Published: 1 November 1996