The geometry of the triple junction between three fluids in equilibrium

TL;DR

This paper analyzes the geometric structure of the triple junction where three fluids in equilibrium meet, establishing monotonicity formulas and cone-shaped blow-up limits to understand the interface configurations.

Contribution

It introduces two new monotonicity formulas at the triple junction and proves that blow-up limits are always conical, advancing the understanding of fluid interface geometry.

Findings

Existence of energy minimizers in bounded variation functions.

Interfaces between two fluids are analytic surfaces.

Blow-up limits at the triple junction are cones.

Abstract

We conduct an analysis of the blow up at the triple junction of three fluids in equilibrium. Energy minimizers have been shown to exist in the class of functions of bounded variations, and the classical theory implies that an interface between two fluids is an analytic surface. We prove two monotonicity formulas at the triple junction for the three-fluid configuration, and show that blow up limits exist and are always cones. We discuss some of the geometric consequences of our results.