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 $ϵ>0$ the solution height must-physically become infinite when $|\gamma –\frac{\pi }{2}|>|{\gamma }_0–ϵ–\frac{\pi }{2}|$, over a subdomain that includes as large a portion of $\Omega$ as desired.