Density estimates for a fluid jet model in the Heisenberg group

In the Heisenberg group setting, we consider the fluid jet functional∫|∇Hnw(ξ)|2+Q(ξ)χ{|w|<1}(ξ)dξ 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 rQ−1rQ−1, while the measure of the exterior medium is of the order of rQrQ, where Q=2(n+1)Q=2(n+1) is the homogeneous dimension. Also, we obtain that |∇Hnu(ξ)|=Q(ξ), in a suitable weak sense, at the free boundary ξ∈∂{|u|<1}ξ∈∂{|u|<1}.