论文标题
最小网络的cojasiewicz-simon不平等：稳定性和收敛性
Łojasiewicz-Simon inequalities for minimal networks: stability and convergence
论文作者
论文摘要
我们通过平面网络的曲率研究了运动的稳定性。我们证明了具有三重连接的平面网络的长度函数的Lojasiewicz-Simon梯度不等式。特别是，这种不等式适用于与$ h^2 $ norm在最小网络中的$ \tfrac23π$相当的连接角度的网络，即边缘也消失的曲率的网络。后者的不等式界限了最小网络的长度$γ_*$与三连接网络的长度之间的凹入力，从上面的$ l^2 $ - 边缘的曲率$γ$ norm $γ$。我们应用此结果来证明最小网络的稳定性是从曲率开始的，从$ h^2 $ norm到达足够接近的网络开始到所有时间都存在并顺利收敛的运动。我们进一步严格构建一个曲率运动的示例，该曲率具有均匀界限的曲率，该曲率在无限的时间内平滑收敛到退化网络。
We investigate stability properties of the motion by curvature of planar networks. We prove Lojasiewicz-Simon gradient inequalities for the length functional of planar networks with triple junctions. In particular, such an inequality holds for networks with junctions forming angles equal to $\tfrac23π$ that are close in $H^2$-norm to minimal networks, i.e., networks whose edges also have vanishing curvature. The latter inequality bounds a concave power of the difference between length of a minimal network $Γ_*$ and length of a triple junctions network $Γ$ from above by the $L^2$-norm of the curvature of the edges of $Γ$. We apply this result to prove the stability of minimal networks in the sense that a motion by curvature starting from a network sufficiently close in $H^2$-norm to a minimal one exists for all times and smoothly converges. We further rigorously construct an example of a motion by curvature having uniformly bounded curvature that smoothly converges to a degenerate network in infinite time.
