  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.
