The L1-relaxed area of the graph of the vortex map: Optimal upper bound

We compute an upper bound for the value of the L1-relaxed area of the graph of the vortex map u: Bl(0) ⊂ℝ2 → ℝ2 u(x):= x/|x|, x ≠ 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.