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arXiv · 2511.01204

Varifold convergence of free boundary Allen--Cahn equation

Jingeon An-Lacroix·Kiichi Tashiro
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Abstract

The free boundary Allen--Cahn equation $\Delta u=0$ 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 $\Gamma$-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.

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Jingeon An-Lacroix, Kiichi Tashiro. 2025-11-03. Varifold convergence of free boundary Allen--Cahn equation. https://arxiv.org/abs/2511.01204

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