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arXiv · 2409.14210

Relaxation of the area of the vortex map: a non-parametric Plateau problem for a catenoid containing a segment

Giovanni Bellettini·Alaa Elshorbagy·Riccardo Scala
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Abstract

Motivated by the study of the non-parametric area $\mathcal A$ of the graph of the vortex map $u$ (a two-codimensional singular surface in $\mathbb R^4$) over the disc $\Omega \subset \mathbb R^2$ of radius $l$, we perform a careful analysis of the singular part of the relaxation of $\mathcal A$ computed at $u$. The precise description is given in terms of a area-minimizing surface in a vertical copy of $\mathbb R^3 \subset \mathbb R^4$, which is a sort of ``catenoid'' containing a segment corresponding to a radius of $\Omega$. The problem involves an area-minimization with a free boundary part; several boundary regularity properties of the minimizer are inspected.

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Giovanni Bellettini, Alaa Elshorbagy, Riccardo Scala. 2024-09-21. Relaxation of the area of the vortex map: a non-parametric Plateau problem for a catenoid containing a segment. https://arxiv.org/abs/2409.14210

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