The isoperimetric inequality for the capillary energy outside convex cylinders

TL;DR

This paper establishes a new isoperimetric inequality for capillary surfaces outside convex cylinders, extending previous results to arbitrary contact angles and characterizing the minimal energy surfaces.

Contribution

It extends the Choe-Ghomi-Ritoré isoperimetric inequality to general contact angles for capillary surfaces outside convex cylinders.

Findings

Capillary energy of surfaces exceeds that of spherical caps unless they are spherical caps themselves.

The result applies to convex cylinders with arbitrary two-dimensional convex sections.

The inequality generalizes known results to all contact angles in (0, π).

Abstract

We study the isoperimetric problem for capillary surfaces with a general contact angle $\theta \in (0, \pi)$, outside convex infinite cylinders with arbitrary two-dimensional convex section. We prove that the capillary energy of any surface supported on any such convex cylinder is strictly larger than that of a spherical cap with the same volume and the same contact angle on a flat support, unless the surface is itself a spherical cap resting on a facet of the cylinder. In this class of convex sets, our result extends for the first time the well-known Choe-Ghomi-Ritor\'e relative isoperimetric inequality, corresponding to the case $\theta = \pi/2$, to general angles.