Second order estimates for a free boundary phase transition

Abstract

It is well known that minimizers of the Allen-Cahn-type functional \[ J_\epsilon(u):=\int_\Omega\frac{\epsilon|\nabla u|^2}{2}+\frac{W(u)}{\epsilon}, \] where $W$ is a double-well potential, resemble minimal surfaces in the sense that their level sets converge to a minimal surface as $\epsilon\to 0$. In this work, we consider the indicator potential $W(\tau)=\chi_{(-1,1)}(\tau)$, which leads to the Bernoulli-type free-boundary problem \[ \left\{\begin{alignedat}{2} \Delta u&=0&\quad&\textrm{in}\quad\{|u|<1\} |\nabla u|&=\epsilon^{-1}&\quad&\textrm{on}\quad\partial \{|u|<1\}. \end{alignedat} \right. \] We provide a short proof that the transition layers are uniformly $C^{2,\alpha}$ regular, up to the free boundary. In addition to the uniform $C^{2,\alpha}$ estimate, we also obtain improved $C^\alpha$ mean curvature bound that decays in an algebraic rate of $\epsilon$, which confirms the convergence of interfaces to the minimal surface in a very strong sense. We present a simple elliptic equation \[ \Delta\phi=H^2-|\mathbf{A}|^2 \] where $\phi=\log(1/|\nabla u|)$ 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.