General rigidity principles for stable and minimal elastic curves
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866910436406853632 |
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| author | Miura, Tatsuya Yoshizawa, Kensuke |
| author_facet | Miura, Tatsuya Yoshizawa, Kensuke |
| contents | For a wide class of curvature energy functionals defined for planar curves under the fixed-length constraint, we obtain optimal necessary conditions for global and local minimizers. Our results extend Maddocks' and Sachkov's rigidity principles for Euler's elastica by a new, unified and geometric approach. This in particular leads to complete classification of stable closed $p$-elasticae for all $p\in(1,\infty)$ and of stable pinned $p$-elasticae for $p\in(1,2]$. Our proof is based on a simple but robust `cut-and-paste' trick without computing the energy nor its second variation, which works well for planar periodic curves but also extends to some non-periodic or non-planar cases. An analytically remarkable point is that our method is directly valid for the highly singular regime $p\in(1,\frac{3}{2}]$ in which the second variation may not exist even for smooth variations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_08384 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | General rigidity principles for stable and minimal elastic curves Miura, Tatsuya Yoshizawa, Kensuke Differential Geometry Analysis of PDEs 49Q10, 53A04 For a wide class of curvature energy functionals defined for planar curves under the fixed-length constraint, we obtain optimal necessary conditions for global and local minimizers. Our results extend Maddocks' and Sachkov's rigidity principles for Euler's elastica by a new, unified and geometric approach. This in particular leads to complete classification of stable closed $p$-elasticae for all $p\in(1,\infty)$ and of stable pinned $p$-elasticae for $p\in(1,2]$. Our proof is based on a simple but robust `cut-and-paste' trick without computing the energy nor its second variation, which works well for planar periodic curves but also extends to some non-periodic or non-planar cases. An analytically remarkable point is that our method is directly valid for the highly singular regime $p\in(1,\frac{3}{2}]$ in which the second variation may not exist even for smooth variations. |
| title | General rigidity principles for stable and minimal elastic curves |
| topic | Differential Geometry Analysis of PDEs 49Q10, 53A04 |
| url | https://arxiv.org/abs/2301.08384 |