arXiv subjects

Jingeon An-Lacroix

Publications and source records attributed to Jingeon An-Lacroix.

4 recordsLinked to original sources

arXiv2025

Parabolic free boundary phase transition and mean curvature flow

It is known that there is a strong relation between the parabolic Allen--Cahn equation and the mean curvature flow, in the sense that the parabolic Allen--Cahn equation can be considered as a "diffused" mean curvature flow. In this work, we derive a forced mean curvature flow \[ v=-H-\partial_ν\log |\nabla u|+f(u)/|\nabla u|, \] satisfied by level surfaces of any solution to the nonlinear parabolic equation \[ \partial_tu=Δu-f(u). \] Moreover, we introduce the notion of the inner gradient flow, and unify parabolic free boundary problems in the gradient flow framework. Finally, we consider the parabolic free boundary Allen--Cahn equation \[ \left\{\begin{alignedat}{2} \partial_tu&=Δu\quad&&\text{in}\quad\{|u|<1\} |\nabla u|&=1/ε\quad&&\text{on}\quad\partial\{|u|<1\}, \end{alignedat} \right. \] and confirm that under reasonable assumptions, the $C^α$ norm of the forcing term $\partial_ν\log|\nabla u|$ converges to zero at an algebraic rate as $ε\to 0$, uniformly in time. This implies that the parabolic free boundary Allen--Cahn equation converges to the mean curvature flow, uniformly (in $ε$ and in time) in the $C^{2,α}$ sense.

math.AP↗

arXiv2025

Varifold convergence of free boundary Allen--Cahn equation

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.

math.AP↗

arXiv2025

Second order estimates for a free boundary phase transition

It is well known that minimizers of the Allen-Cahn-type functional \[ J_ε(u):=\int_Ω\frac{ε|\nabla u|^2}{2}+\frac{W(u)}ε, \] where $W$ is a double-well potential, resemble minimal surfaces in the sense that their level sets converge to a minimal surface as $ε\to 0$. In this work, we consider the indicator potential $W(τ)=χ_{(-1,1)}(τ)$, which leads to the Bernoulli-type free-boundary problem \[ \left\{\begin{alignedat}{2} Δu&=0&\quad&\textrm{in}\quad\{|u|<1\} |\nabla u|&=ε^{-1}&\quad&\textrm{on}\quad\partial \{|u|<1\}. \end{alignedat} \right. \] We provide a short proof that the transition layers are uniformly $C^{2,α}$ regular, up to the free boundary. In addition to the uniform $C^{2,α}$ estimate, we also obtain improved $C^α$ mean curvature bound that decays in an algebraic rate of $ε$, which confirms the convergence of interfaces to the minimal surface in a very strong sense. We present a simple elliptic equation \[ Δϕ=H^2-|\mathbf{A}|^2 \] where $ϕ=\log(1/|\nabla u|)$ is the log-gradient of $u$, $H$ and $\mathbf{A}$ are the mean curvature and the second fundamental form of level surfaces, respectively. From this, the uniform estimates readily follow. The whole argument is performed in a general Riemannian manifold setting.

math.AP↗

arXiv2025

Convergence of the fractional Yamabe flow for arbitrary initial energy

Since the seminal paper of Graham and Zworski (Invent. Math. 2003), conformal geometric problems are studied in the fractional setting. We consider the convergence of fractional Yamabe flow, which is previously known under small initial energy assumption. Inspired by the deep work of Brendle (J. Diff. Geom. 2005), we obtain the full convergence result for arbitrary initial energy, whenever the (fractional) positive mass conjecture is valid.

math.AP↗