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

Abstract

We compute the value of the $L^1$-relaxed area of the graph of the map $u:B_l(0)\subset {\mathbb{R}}^2\mapsto {\mathbb{R}}^2$, $u(x):=x/|x|$, $x\ne 0$, for all values of $l>0$. 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.