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