A fractional notion of length and an associated nonlocal curvature
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2018
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| _version_ | 1866910170657849344 |
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| author | Seguin, Brian |
| author_facet | Seguin, Brian |
| contents | Here a new notion of fractional length of a smooth curve, which depends on a parameter $σ$, is introduced that is analogous to the fractional perimeter functional of sets that has been studied in recent years. It is shown that in an appropriate limit the fractional length converges to the traditional notion of length up to a multiplicative constant. Since a curve that connects two points of minimal length must have zero curvature, the Euler--Lagrange equation associated with the fractional length is used to motivate a nonlocal notion of curvature for a curve. This is analogous to how the fractional perimeter has been used to define a nonlocal mean curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1808_08654 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | A fractional notion of length and an associated nonlocal curvature Seguin, Brian Differential Geometry 53A04, 28A75, 49Q15 Here a new notion of fractional length of a smooth curve, which depends on a parameter $σ$, is introduced that is analogous to the fractional perimeter functional of sets that has been studied in recent years. It is shown that in an appropriate limit the fractional length converges to the traditional notion of length up to a multiplicative constant. Since a curve that connects two points of minimal length must have zero curvature, the Euler--Lagrange equation associated with the fractional length is used to motivate a nonlocal notion of curvature for a curve. This is analogous to how the fractional perimeter has been used to define a nonlocal mean curvature. |
| title | A fractional notion of length and an associated nonlocal curvature |
| topic | Differential Geometry 53A04, 28A75, 49Q15 |
| url | https://arxiv.org/abs/1808.08654 |