The $L^1$-relaxed area of the graph of the vortex map

We compute the value of the $L^1$-relaxed area of the graph of the map $u : B_l(0)\subset \mathbb R^2 \mapsto \mathbb R^2$,$u(x):= x/\vert x\vert$,$x \neq 0$,forallvaluesof$l>0$\mathrm{.}Interestingly,for$l$ in a certain range, in particular $l$ not too large, a Plateau-type problem, having as solution a sort of catenoid constrained to contain a segment, has to be solved.