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

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.

arXiv admin note: substantial text overlap with arXiv:2107.07236