Phase field approximation for Plateau's problem: a curve geodesic distance penalty approach
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909662983487488 |
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| author | Bonnivard, Matthieu Bretin, Elie Lemenant, Antoine Machefert, Eve |
| author_facet | Bonnivard, Matthieu Bretin, Elie Lemenant, Antoine Machefert, Eve |
| contents | This work focuses on a phase field approximation of Plateau's problem. Inspired by Reifenberg's point of view, we introduce a model that combines the Ambrosio-Torterelli energy with a geodesic distance term, which can be considered as a generalization of the approach developed by Bonnivard, Lemenant and Santambrogio to approximate solutions to Steiner's problem. First, we present a Gamma-convergence analysis of this model in the simple case of a single curve located on the edge of a cylinder. In a numerical section, we detail the numerical optimisation schemes used to minimize this energy for numerous examples, for which good approximations of solutions to Plateau's problem are found. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22273 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Phase field approximation for Plateau's problem: a curve geodesic distance penalty approach Bonnivard, Matthieu Bretin, Elie Lemenant, Antoine Machefert, Eve Optimization and Control Numerical Analysis Analysis of PDEs This work focuses on a phase field approximation of Plateau's problem. Inspired by Reifenberg's point of view, we introduce a model that combines the Ambrosio-Torterelli energy with a geodesic distance term, which can be considered as a generalization of the approach developed by Bonnivard, Lemenant and Santambrogio to approximate solutions to Steiner's problem. First, we present a Gamma-convergence analysis of this model in the simple case of a single curve located on the edge of a cylinder. In a numerical section, we detail the numerical optimisation schemes used to minimize this energy for numerous examples, for which good approximations of solutions to Plateau's problem are found. |
| title | Phase field approximation for Plateau's problem: a curve geodesic distance penalty approach |
| topic | Optimization and Control Numerical Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2506.22273 |