A two-point function approach to connectedness of drops in convex potentials

TL;DR

This paper proves the convexity and connectedness of volume-constrained minimizers in energies with surface tension and convex potentials, introducing a novel two-point function to analyze convexity properties.

Contribution

It introduces a new two-point function method to establish convexity and connectedness of minimizers, solving an old open problem in the field.

Findings

Minimizers are connected under volume constraints.

Minimizers are convex in two dimensions.

Convexity holds without volume constraints.

Abstract

We establish connectedness of volume constrained minimisers of energies involving surface tensions and convex potentials. By a previous result of McCann, this implies that minimisers are convex in dimension two. This positively answers an old question of Almgren. We also prove convexity of minimisers when the volume constraint is dropped. Our proof is based on the introduction of a new "two-point function" which measures the lack of convexity and which gives rise to a negative second variation of the energy.