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arXiv · 2409.18143

The $L^1$-relaxed area of the graph of the vortex map: optimal upper bound

Giovanni Bellettini·Alaa Elshorbagy·Riccardo Scala
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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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Giovanni Bellettini, Alaa Elshorbagy, Riccardo Scala. 2024-09-21. The $L^1$-relaxed area of the graph of the vortex map: optimal upper bound. https://arxiv.org/abs/2409.18143

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