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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2511.01204 |
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| _version_ | 1866918182339477504 |
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| author | An, Jingeon Tashiro, Kiichi |
| author_facet | An, Jingeon Tashiro, Kiichi |
| contents | The free boundary Allen--Cahn equation $Δu=0$ in $\{|u|<1\}$, $|\nabla u|=1/\varepsilon$ on $\partial\{|u|<1\}$ has recently attracted considerable attention because it retains the essential features of the classical Allen--Cahn equation while being significantly more tractable. In this work, we establish the free boundary analogue of the seminal Hutchinson--Tonegawa theory, developing the varifold convergence framework for solutions of the free boundary Allen--Cahn equation to minimal surfaces. In addition, we provide the $Γ$-convergence of the free boundary Allen--Cahn energy to the area functional, and the conservation of local minimization property. This foundation is expected to be used in further applications of the free boundary Allen--Cahn equation in the study of minimal surfaces, such as providing an alternative proof of celebrated Yau's conjecture, possibly with simpler and more complete arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01204 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Varifold convergence of free boundary Allen--Cahn equation An, Jingeon Tashiro, Kiichi Analysis of PDEs 35R35, 35N25, 53A10 The free boundary Allen--Cahn equation $Δu=0$ in $\{|u|<1\}$, $|\nabla u|=1/\varepsilon$ on $\partial\{|u|<1\}$ has recently attracted considerable attention because it retains the essential features of the classical Allen--Cahn equation while being significantly more tractable. In this work, we establish the free boundary analogue of the seminal Hutchinson--Tonegawa theory, developing the varifold convergence framework for solutions of the free boundary Allen--Cahn equation to minimal surfaces. In addition, we provide the $Γ$-convergence of the free boundary Allen--Cahn energy to the area functional, and the conservation of local minimization property. This foundation is expected to be used in further applications of the free boundary Allen--Cahn equation in the study of minimal surfaces, such as providing an alternative proof of celebrated Yau's conjecture, possibly with simpler and more complete arguments. |
| title | Varifold convergence of free boundary Allen--Cahn equation |
| topic | Analysis of PDEs 35R35, 35N25, 53A10 |
| url | https://arxiv.org/abs/2511.01204 |