The $L^1$-relaxed area of the graph of the vortex map: optimal upper bound
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913520668377088 |
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| author | Bellettini, Giovanni Elshorbagy, Alaa Scala, Riccardo |
| author_facet | Bellettini, Giovanni Elshorbagy, Alaa Scala, Riccardo |
| contents | We compute an upper bound for the value of the $L^1$-relaxed area of the graph of the vortex map $u : B_l(0)\subset \mathbb R^2 \to \mathbb R^2$, $u(x):= x/\vert x\vert$, $x \neq 0$, for all values of $l>0$. Together with a previously proven lower bound, this upper bound turns out to be optimal. Interestingly, for the radius $l$ in a certain range, in particular $l$ not too large, a Plateau-type problem, having as solution a sort of catenoid constrained to contain a segment, has to be solved. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_18143 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $L^1$-relaxed area of the graph of the vortex map: optimal upper bound Bellettini, Giovanni Elshorbagy, Alaa Scala, Riccardo Analysis of PDEs 49Q15, 49Q20, 49J45, 58E12 We compute an upper bound for the value of the $L^1$-relaxed area of the graph of the vortex map $u : B_l(0)\subset \mathbb R^2 \to \mathbb R^2$, $u(x):= x/\vert x\vert$, $x \neq 0$, for all values of $l>0$. Together with a previously proven lower bound, this upper bound turns out to be optimal. Interestingly, for the radius $l$ in a certain range, in particular $l$ not too large, a Plateau-type problem, having as solution a sort of catenoid constrained to contain a segment, has to be solved. |
| title | The $L^1$-relaxed area of the graph of the vortex map: optimal upper bound |
| topic | Analysis of PDEs 49Q15, 49Q20, 49J45, 58E12 |
| url | https://arxiv.org/abs/2409.18143 |