The Modified Canonical Proboscis

Abstract

A canonical proboscis domain $\Omega$ corresponding to contact angle as introduced $\gamma_0$ by Fischer and Finn and later studied by Finn and Leise, has the property that a solution of the capillary problem exists in $\Omega$ for contact angle $\gamma$ if and only if $| \gamma – \frac{\pi}{2}| < | \gamma_0 – \frac{\pi}{2}|$. We show in this paper that every such domain can be modified so as to yield the existence of a bounded solution also at the angle $\gamma_0$. The modification can be effected in such a way that for prescribed $\epsilon > 0$ the solution height must-physically become infinite when $| \gamma – \frac{\pi}{2}| > |\gamma_0 – \epsilon – \frac{\pi}{2}|$, over a subdomain that includes as large a portion of $\Omega$ as desired.