Density estimates for a fluid jet model in the Heisenberg group

In the Heisenberg group setting, we consider the fluid jet functional ∫ | ∇ H n w ( ξ ) | 2 + Q ( ξ ) χ { | w | < 1 } ( ξ ) d ξ and we prove some density estimates for local minima. In particular, for large r, the measure of the minimal jet in balls of radius r centered at the jet is of the order of r Q − 1 , while the measure of the exterior medium is of the order of r Q , where Q = 2 ( n + 1 ) is the homogeneous dimension. Also, we obtain that | ∇ H n u ( ξ ) | = Q ( ξ ) , in a suitable weak sense, at the free boundary ξ ∈ ∂ { | u | < 1 } .