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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2406.14788 |
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| _version_ | 1866929557816213504 |
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| author | Patrizi, Stefania Vaughan, Mary |
| author_facet | Patrizi, Stefania Vaughan, Mary |
| contents | We prove that the mean curvature of a smooth surface in $\mathbb{R}^n$, $n\geq 2$, arises as the limit of a sequence of functions that are intrinsically related to the difference between an $n$- and $1$-dimensional fractional Laplacian of a phase transition. Depending on the order of the fractional Laplace operator, we recover the fractional mean curvature or the classical mean curvature of the surface. Moreover, we show that this is an essential ingredient for deriving the evolution of fronts in fractional reaction-diffusion equations such as those for atomic dislocations in crystals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14788 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A convergence result for the derivation of front propagation in nonlocal phase field models Patrizi, Stefania Vaughan, Mary Analysis of PDEs 35R09, 74N20, 53E10, 35R11, 47G20 We prove that the mean curvature of a smooth surface in $\mathbb{R}^n$, $n\geq 2$, arises as the limit of a sequence of functions that are intrinsically related to the difference between an $n$- and $1$-dimensional fractional Laplacian of a phase transition. Depending on the order of the fractional Laplace operator, we recover the fractional mean curvature or the classical mean curvature of the surface. Moreover, we show that this is an essential ingredient for deriving the evolution of fronts in fractional reaction-diffusion equations such as those for atomic dislocations in crystals. |
| title | A convergence result for the derivation of front propagation in nonlocal phase field models |
| topic | Analysis of PDEs 35R09, 74N20, 53E10, 35R11, 47G20 |
| url | https://arxiv.org/abs/2406.14788 |