Varifold convergence of free boundary Allen--Cahn equation

The free boundary Allen–Cahn equation $\mathrm{\Delta }u=0$ in $\{\mathrm{\mid }u\mathrm{\mid }<1\}$, $\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }=1\mathrm{/}\epsilon$ on $\mathrm{\partial }\{\mathrm{\mid }u\mathrm{\mid }<1\}$ 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 $\mathrm{\Gamma }$-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.