*Preprint*

**Inserted:** 21 sep 2024

**Last Updated:** 21 sep 2024

**Year:** 2024

**Abstract:**

We prove a lower bound for the value of the $L^1$-relaxed area of the graph of the map $u : B_l(0) \setminus \{0\}\subset \mathbb R^2 \to \mathbb R^2$, $u(x):= x/\vert x\vert$, $x \neq 0$, 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.

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