Preprint

Inserted: 21 sep 2024
Last Updated: 21 sep 2024

Year: 2024

Abstract:

We compute an upper bound for the value of the $L^1$-relaxed area of the graph of the vortex map $u : B_l(0)\subset \mathbb R^2 \to \mathbb R^2$, $u(x):= x/\vert x\vert$, $x \neq 0$, 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.

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• bellettini_scala_upper_bound.pdf