On the three-dimensional shape of a crystal
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929369275957248 |
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| author | Indrei, Emanuel Karakhanyan, Aram |
| author_facet | Indrei, Emanuel Karakhanyan, Aram |
| contents | In this paper we completely settle the Almgren problem in $\mathbb R^3$ under some generic conditions on the potential and tension functions. The problem, among other things, appears in classical thermodynamics when one is to understand if minimizing the free energy with convex potential and under a mass constraint generates a convex crystal. Our new idea in proving a three-dimensional convexity theorem is to utilize a stability theorem when $m$ is small, convexity when $m$ is small, and the first variation PDE with a new maximum principle approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00241 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the three-dimensional shape of a crystal Indrei, Emanuel Karakhanyan, Aram Analysis of PDEs Mathematical Physics Differential Geometry 49Jxx, 49Sxx, 49Q20, 52Axx, 80Axx, 82B30, 28A75 In this paper we completely settle the Almgren problem in $\mathbb R^3$ under some generic conditions on the potential and tension functions. The problem, among other things, appears in classical thermodynamics when one is to understand if minimizing the free energy with convex potential and under a mass constraint generates a convex crystal. Our new idea in proving a three-dimensional convexity theorem is to utilize a stability theorem when $m$ is small, convexity when $m$ is small, and the first variation PDE with a new maximum principle approach. |
| title | On the three-dimensional shape of a crystal |
| topic | Analysis of PDEs Mathematical Physics Differential Geometry 49Jxx, 49Sxx, 49Q20, 52Axx, 80Axx, 82B30, 28A75 |
| url | https://arxiv.org/abs/2406.00241 |