The modified canonical proboscis (Q1909645)

Summary: A canonical proboscis domain \(\Omega\) corresponding to contact angle \(\gamma_0\) has the property that a solution of the capillary problem exists in \(\Omega\) for contact angle \(\gamma\) if and only if \(|\gamma- {\pi\over 2}|< |\gamma_0- {\pi\over 2}|\). We show 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 \(\varepsilon>0\) the solution height must physically become infinite when \(|\gamma- {\pi\over 2}|> |\gamma_0- \varepsilon- {\pi\over 2}|\), over a subdomain that includes as large a portion of \(\Omega\) as desired.