论文标题
Brakke的规律性用于Allen-Cahn流
Brakke Regularity for the Allen-Cahn Flow
论文作者
论文摘要
在本文中，我们证明了Brakke的$ \ varepsilon $ - 型定理的抛物线allen-cahn方程定理。特别是，我们显示出均匀的$ c^{2，α} $的过渡层的规律性，融合到平滑平均曲率流量为$ \ varepsilon \ rightarrow0 $。该证明利用了单调性公式，抛物线式Lipschitz近似和爆炸的艾伦·卡恩（Allen-Cahn）版本。还获得了抛物线代理艾伦-CAHN的整个永恒溶液的相应差距定理。作为规律定理的应用，我们对伊尔曼宁的问题给出了肯定的答案，即在平均凸设置中，$ \ mathbf {bv} $收敛中没有取消。
In this paper we prove an analogue of the Brakke's $\varepsilon$-regularity theorem for the parabolic Allen-Cahn equation. In particular, we show uniform $C^{2,α}$ regularity for the transition layers converging to smooth mean curvature flows as $\varepsilon\rightarrow0$. The proof utilises Allen-Cahn versions of the monotonicity formula, parabolic Lipschitz approximation and blowups. A corresponding gap theorem for entire eternal solutions of the parabolic Allen-Cahn is also obtained. As an application of the regularity theorem, we give an affirmative answer to a question of Ilmanen that there is no cancellation in $\mathbf {BV}$ convergence in the mean convex setting.
