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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.03810 |
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| _version_ | 1866908481822392320 |
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| author | An, Jingeon |
| author_facet | An, Jingeon |
| contents | 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 $ε\rightarrow 0$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03810 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Second order estimates for a free boundary phase transition An, Jingeon Analysis of PDEs 35R35 (Free boundary problems for PDEs) 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 $ε\rightarrow 0$. 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. |
| title | Second order estimates for a free boundary phase transition |
| topic | Analysis of PDEs 35R35 (Free boundary problems for PDEs) |
| url | https://arxiv.org/abs/2507.03810 |