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

Abstract

We prove a lower bound for the value of the $L^1$-relaxed area of the graph of the map $u:B_l\left(0\right)\setminus \left\{0\right\}\subset R^2\to R^2$, $u\left(x\right):=x/\left|x\right|$, $x\ne 0$, for all values of the radius $l>0$. In the computation of the singular part of the relaxed area, for $l$ in a certain range, in particular $l$ not too large, a nonparametric Plateau-type problem with partial free boundary has to be solved. Our lower bound turns out to be optimal, in view of an upper bound proven in a companion paper.