论文标题
过渡层的二阶估计值和抛物线的曲率估计值
Second order estimates for transition layers and a curvature estimate for the parabolic Allen-Cahn
论文作者
论文摘要
抛物线的艾伦-CAHN方程是一个半线性的部分微分方程，该方程与平均曲率流相关联。我们显示了抛物线齿轮方程到平均曲率流的改善的收敛性，这是椭圆形艾伦 - 卡恩（Allen-Cahn）改善的抛物线类似物对Wang-Wei和Chodosh-Mantoulidis的最小表面的抛物线类似物。更确切地说，我们显示了相变级集合集合在$ c^2 $中，然后它们收敛于$ c^{2，θ} $。作为应用程序，我们获得了抛物线抗抛物线方程的曲率估计值，可以将其视为Brakke's和White的规律定理的扩散版本，用于平均曲率流量
The parabolic Allen-Cahn equation is a semilinear partial differential equation linked to the mean curvature flow by a singular perturbation. We show an improved convergence property of the parabolic Allen-Cahn equation to the mean curvature flow, which is the parabolic analogue of the improved convergence property of the elliptic Allen-Cahn to minimal surfaces by Wang-Wei and Chodosh-Mantoulidis. More precisely, we show if the phase-transition level sets are converging in $C^2$, then they converge in $C^{2,θ}$. As an application, we obtain a curvature estimate for parabolic Allen-Cahn equation, which can be viewed as a diffused version of Brakke's and White's regularity theorem for mean curvature flow
