Gradient flow of phase transitions with fixed contact angle
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916934084198400 |
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| author | Marshall-Stevens, Kobe Takada, Mayu Tonegawa, Yoshihiro Workman, Myles |
| author_facet | Marshall-Stevens, Kobe Takada, Mayu Tonegawa, Yoshihiro Workman, Myles |
| contents | We study the gradient flow of the Allen-Cahn equation with fixed boundary contact angle in Euclidean domains for initial data with bounded energy. Under general assumptions, we establish both interior and boundary convergence properties for the solutions and associated energy measures. Under various boundary non-concentration assumptions, we show that, for almost every time, the associated limiting varifolds satisfy generalised contact angle conditions and have bounded first variation, as well as deducing that the trace of the limit of the solutions coincides with the limit of their traces. Moreover, we derive an Ilmanen type monotonicity formula, for initial data with bounded energy, valid for the associated energy measures up to the boundary. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_17979 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gradient flow of phase transitions with fixed contact angle Marshall-Stevens, Kobe Takada, Mayu Tonegawa, Yoshihiro Workman, Myles Analysis of PDEs Differential Geometry 53E10, 49Q20, 35K20 We study the gradient flow of the Allen-Cahn equation with fixed boundary contact angle in Euclidean domains for initial data with bounded energy. Under general assumptions, we establish both interior and boundary convergence properties for the solutions and associated energy measures. Under various boundary non-concentration assumptions, we show that, for almost every time, the associated limiting varifolds satisfy generalised contact angle conditions and have bounded first variation, as well as deducing that the trace of the limit of the solutions coincides with the limit of their traces. Moreover, we derive an Ilmanen type monotonicity formula, for initial data with bounded energy, valid for the associated energy measures up to the boundary. |
| title | Gradient flow of phase transitions with fixed contact angle |
| topic | Analysis of PDEs Differential Geometry 53E10, 49Q20, 35K20 |
| url | https://arxiv.org/abs/2411.17979 |