Minimizing cones for fractional capillarity problems

TL;DR

This paper studies fractional capillarity energy, establishing boundary monotonicity formulas and showing that in 2D, the only minimizer is the fractional Young's law cone, advancing understanding of fractional minimal surfaces.

Contribution

Introduces a fractional capillarity energy model, derives a boundary monotonicity formula, and classifies minimizers in the planar case as a fractional Young's law cone.

Findings

Blow-up limits of minimizers converge to cones.

In 2D, the only minimizer is the fractional Young's law cone.

Established a boundary monotonicity formula for fractional capillarity.

Abstract

We consider a fractional version of Gauss capillarity energy. A suitable extension problem is introduced to derive a boundary monotonicity formula for local minimizers of this fractional capillarity energy. As a consequence, blow-up limits of local minimizers are shown to subsequentially converge to minimizing cones. Finally, we show that in the planar case there is only one possible fractional minimizing cone, the one determined by the fractional version of Young's law.