Stationary curves under the Möbius-Plateau energy
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913725825417216 |
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| author | Lipton, Max Nair, Gokul |
| author_facet | Lipton, Max Nair, Gokul |
| contents | Plateau problems with elastic boundary energies have been of recent theoretical and applied interest. However, strong assumptions have to be made to avoid self-intersections of the boundary curve during energy minimization. We introduce a class of Plateau problems for boundaries with self-repulsive energies that obviates self-contact in energy minimization problems. For the self-repulsive energy, we choose the Möbius Energy introduced by O'Hara due to its myriad regularity properties shown by Freedman et al. We first prove an existence theorem for this Möbius-Plateau problem in the class of closed Lipschitz curves of a given irreducible knot-type spanned by immersed discs. We then turn our attention to Möbius-Plateau variations of helicoidal strips, which are classified as "screw-like" or "ribbon-like" based on the signs of the radii of the boundary helices. By analyzing the Euler-Lagrange equations, we show that screw-like solutions are plentiful, whilst ribbon-like solutions impose strong constraints on their parameters: they must have high frequency (equivalently, low pitch), thin width in comparison to the frequency, and remain close to the axis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_12678 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Stationary curves under the Möbius-Plateau energy Lipton, Max Nair, Gokul Differential Geometry Analysis of PDEs Plateau problems with elastic boundary energies have been of recent theoretical and applied interest. However, strong assumptions have to be made to avoid self-intersections of the boundary curve during energy minimization. We introduce a class of Plateau problems for boundaries with self-repulsive energies that obviates self-contact in energy minimization problems. For the self-repulsive energy, we choose the Möbius Energy introduced by O'Hara due to its myriad regularity properties shown by Freedman et al. We first prove an existence theorem for this Möbius-Plateau problem in the class of closed Lipschitz curves of a given irreducible knot-type spanned by immersed discs. We then turn our attention to Möbius-Plateau variations of helicoidal strips, which are classified as "screw-like" or "ribbon-like" based on the signs of the radii of the boundary helices. By analyzing the Euler-Lagrange equations, we show that screw-like solutions are plentiful, whilst ribbon-like solutions impose strong constraints on their parameters: they must have high frequency (equivalently, low pitch), thin width in comparison to the frequency, and remain close to the axis. |
| title | Stationary curves under the Möbius-Plateau energy |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2208.12678 |