We consider the sharp interface limit of the Allen-Cahn equation with homogeneous Neumann boundary condition in a two-dimensional domain $\Omega$, in the situation where an interface has developed and intersects $\partial\Omega$. Here a parameter $\varepsilon>0$ in the equation, which is related to the thickness of the diffuse interface, is sent to zero. The limit problem is given by mean curvature flow with a $90$\textdegree-contact angle condition and convergence using strong norms is shown for small times. Here we assume that a smooth solution to this limit problem exists on $[0,T]$ for some $T>0$ and that it can be parametrized suitably. With the aid of asymptotic expansions we construct an approximate solution for the Allen-Cahn equation and estimate the difference of the exact and approximate solution with the aid of a spectral estimate for the linearized Allen-Cahn operator.

中文翻译：

---

Allen-Cahn 方程收敛到 2D 中接触角为 90o 的平均曲率流

我们考虑在二维域 $\Omega$ 中具有齐次 Neumann 边界条件的 Allen-Cahn 方程的锐界面极限，在界面已经发展并与 $\partial\Omega$ 相交的情况下。这里方程中与漫反射界面厚度有关的参数$\varepsilon>0$被置为0。极限问题由具有 $90$\textdegree-contact angle 条件的平均曲率流给出，并且使用强范数的收敛性显示为小时间。在这里，我们假设在 $[0,T]$ 上存在对某些 $T>0$ 的限制问题的平滑解，并且可以适当地对其进行参数化。