Discussion of a uniqueness result in "Equilibrium Configurations for a Floating Drop"

TL;DR

This paper critiques a previous uniqueness theorem for unbounded liquid bridges, corrects its proof with additional hypotheses, and employs spectral methods to numerically verify one of these hypotheses.

Contribution

It identifies errors in a prior proof, refines the theorem with new hypotheses, and develops a spectral numerical method to support the corrected result.

Findings

The original proof was incorrect without additional hypotheses.

Theorem holds under the added three hypotheses.

Numerical methods support the validity of one hypothesis.

Abstract

We analyze a uniqueness result presented by Elcrat, Neel, and Siegel \cite{ENS} for unbounded liquid bridges, and show that the proof they presented is incorrect. We add a choice of three hypothesis to their stated theorem and show that their result holds under this condition. Then we use Chebyshev spectral methods to build a numerical method to approximate solutions to a related boundary value problems that show one of the three hypothesis holds.