论文标题
不均匀的艾伦 - cahn方程和规定的均值狂热性超曲面的存在
The inhomogeneous Allen--Cahn equation and the existence of prescribed-mean-curvature hypersurfaces
论文作者
论文摘要
我们证明，对于任何给定的紧凑型riemannian歧管$ n $的尺寸$ n+1 \ geq 3 $和任何非负Lipschitz在$ n $上的功能$ g $，存在一个Quasi-embedded，bydershypersurface $ m \ m \ m \ m \ subset n，$ c^class $ c^{2，$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ in for not in y nimage nimage nimage nimage nigral nigral nigration nigral nigration nigration nigral nigres at（0.11双面沉浸式的平均曲率由$gν$给出，以适应连续的单位正常$ν$。此外，如果$ 2 \ 2 \ leq n \ leq 6，如果$ n = 7 $且满足$ {\ Mathcal H}^{n-7 +γ}（n-7 +γ}（σ）= 0 $ 0 $ geq 0 $ n，则单数集合$σ= \ overline {m} \ setMinus m $是空的。在这里，quasi插入的意味着，在每个非插入点附近，$ m $是两个嵌入式$ c^{2，α} $磁盘的结合，与每个磁盘均位于另一侧的每个磁盘上，分为切线。如果$ g> 0 $，则$ \ overline {m} $是caccioppoli集的边界。我们对此定理的证明是理论的，并且依赖于$ g> 0 $和$ g \ in c^{1,1}（n）$，（i）（i）（i）（i）山间通道的解决方案构建解决方案的山通构建，以使整体范围内的整体范围界定的整体序列（II）的定期结果（ii）的定期范围，涉及莫尔斯（Morse）的序列界面，范围有界面的界面，范围有界面，这些杂物（艾伦 - cahn方程。基于我们建立的估计，近似值的非负Lipschitz $ g $的情况下。
We prove that for any given compact Riemannian manifold $N$ of dimension $n+1 \geq 3$ and any non-negative Lipschitz function $g$ on $N$, there exists a quasi-embedded, boundaryless hypersurface $M \subset N,$ of class $C^{2, α}$ for any $α\in (0,1),$ such that $M$ is the image of a two-sided immersion whose mean curvature is given by $gν$ for an appropriate choice of continuous unit normal $ν$ to the immersion; and moreover, the singular set $Σ= \overline{M} \setminus M$ is empty if $2 \leq n \leq 6,$ finite if $n=7$ and satisfies ${\mathcal H}^{n-7 + γ}(Σ) = 0$ for every $γ>0$ if $n \geq 8$. Here quasi-embedded means that near every non-embedded point, $M$ is the union of two embedded $C^{2, α}$ disks intersecting tangentially with each disk lying on one side of the other. If $g >0$ then $\overline{M}$ is the boundary of a Caccioppoli set. Our proof of this theorem is PDE theoretic and relies, when $g>0$ and $g\in C^{1,1}(N)$, on (i) a mountain pass construction of solutions to the inhomogeneous Allen--Cahn equation and (ii) a regularity result for integral varifolds arising from a Morse-index bounded, energy bounded, sequence of solutions to the (inhomogeneous) Allen--Cahn equation. The case of non-negative Lipschitz $g$ follows by approximation, based on the estimates that we establish.
