The $L^1$-relaxed area of the graph of the vortex map: optimal upper bound

We compute an upper bound for the value of the $L^1$-relaxed area of the graph of the vortex map $u:B_l(0)\subset {\mathbb{R}}^2\to {\mathbb{R}}^2$, $u(x):=x\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$, $x\mathrm{\ne }0$, for all values of $l>0$. Together with a previously proven lower bound, this upper bound turns out to be optimal. Interestingly, for the radius $l$ 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.