Varifold convergence of free boundary Allen--Cahn equation

TL;DR

This paper develops a varifold convergence framework for the free boundary Allen--Cahn equation, linking solutions to minimal surfaces and establishing foundational results for future geometric analysis applications.

Contribution

It extends Hutchinson--Tonegawa theory to free boundary problems, proving varifold convergence, $ ext{Gamma}$-convergence, and conservation of minimality for solutions.

Findings

Established varifold convergence of free boundary Allen--Cahn solutions to minimal surfaces.

Proved $ ext{Gamma}$-convergence of the energy to the area functional.

Demonstrated conservation of local minimization properties.

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.