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

2024/09/21 by Giovanni Bellettini, Bellettini, Giovanni, Alaa Elshorbagy +3 · 1 citation
Earth and Planetary Sciences · Mathematics · #49J45 #49Q15 #49Q20 #58E12 #Algebraic and Geometric Analysis #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geophysics and Gravity Measurements

paper · pdf · doi:10.48550/arxiv.2409.14210

openalex publication_date 2024/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 R⁴) over the disc Ω⊂ \mathbb R² 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 R³ ⊂ \mathbb R⁴, 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.

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