Relaxation of the area of the vortex map: a non-parametric Plateau problem for a catenoid containing a segment
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| Format: | Preprint |
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2024
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| _version_ | 1866912039804338176 |
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| author | Bellettini, Giovanni Elshorbagy, Alaa Scala, Riccardo |
| author_facet | Bellettini, Giovanni Elshorbagy, Alaa Scala, Riccardo |
| contents | 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 $Ω\subset \mathbb R^2$ 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 $Ω$. The problem involves an area-minimization with a free boundary part; several boundary regularity properties of the minimizer are inspected. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_14210 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relaxation of the area of the vortex map: a non-parametric Plateau problem for a catenoid containing a segment Bellettini, Giovanni Elshorbagy, Alaa Scala, Riccardo Analysis of PDEs 49Q15, 49Q20, 49J45, 58E12 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 $Ω\subset \mathbb R^2$ 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 $Ω$. The problem involves an area-minimization with a free boundary part; several boundary regularity properties of the minimizer are inspected. |
| title | Relaxation of the area of the vortex map: a non-parametric Plateau problem for a catenoid containing a segment |
| topic | Analysis of PDEs 49Q15, 49Q20, 49J45, 58E12 |
| url | https://arxiv.org/abs/2409.14210 |