Plateau flow or the heat flow for half-harmonic maps
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866914803406077952 |
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| author | Struwe, Michael |
| author_facet | Struwe, Michael |
| contents | Using the interpretation of the half-Laplacian on $S^1$ as the Dirichlet-to-Neumann operator for the Laplace equation on the ball $B$, we devise a classical approach to the heat flow for half-harmonic maps from $S^1$ to a closed target manifold $N$, recently studied by Wettstein, and for arbitrary finite-energy data we obtain a result fully analogous to the author's 1985 results for the harmonic map heat flow of surfaces and in similar generality. When $N$ is a smoothly embedded, oriented closed curve $Γ$ the half-harmonic map heat flow may be viewed as an alternative gradient flow for a variant of the Plateau problem of disc-type minimal surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_02083 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Plateau flow or the heat flow for half-harmonic maps Struwe, Michael Differential Geometry Analysis of PDEs 35K55, 35K65, 53E99 Using the interpretation of the half-Laplacian on $S^1$ as the Dirichlet-to-Neumann operator for the Laplace equation on the ball $B$, we devise a classical approach to the heat flow for half-harmonic maps from $S^1$ to a closed target manifold $N$, recently studied by Wettstein, and for arbitrary finite-energy data we obtain a result fully analogous to the author's 1985 results for the harmonic map heat flow of surfaces and in similar generality. When $N$ is a smoothly embedded, oriented closed curve $Γ$ the half-harmonic map heat flow may be viewed as an alternative gradient flow for a variant of the Plateau problem of disc-type minimal surfaces. |
| title | Plateau flow or the heat flow for half-harmonic maps |
| topic | Differential Geometry Analysis of PDEs 35K55, 35K65, 53E99 |
| url | https://arxiv.org/abs/2202.02083 |