Varifold convergence of free boundary Allen--Cahn equation

TL;DR

This work develops a free boundary analogue of the Hutchinson–Tonegawa varifold convergence framework for the Allen–Cahn equation. It proves that, under a uniform energy bound, solutions u_ε converge to a BV limit u_0 taking values ±1 and that the associated energy measures converge to an (n−1)-rectifiable, stationary varifold V, with Γ-convergence of the free boundary energy J_ε to the area functional J_0. In addition, the paper shows that, under a uniform C^2 bound, the limit varifold is integral and carries a density with a precise parity structure, reflecting the free boundary geometry. The results lay the foundation for applying free boundary Allen–Cahn methods to minimal surface theory, including potential alternate proofs of classical conjectures, and establish a robust variational framework for the free boundary transition layers.

Abstract

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.

Theorems & Definitions (33)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3: $\Gamma$-convergence
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Proposition 2.5
• Lemma 3.1
• proof