论文标题
正常网络的曲率的存在和运动的独特性
Existence and Uniqueness of the Motion by Curvature of regular networks
论文作者
论文摘要
当初始基准属于$ w^{2- \ frac {2} {2} {p}} _ p $时，我们证明了通过$ \ mathbb {r}^n $在$ \ mathbb {r}^n $中的运动的存在和独特性，与三重交界处，与单位相切的曲线相切的向量形成costrring curves Angles Angles Angles of Cunves of Bugves of Cunves of Cyves of Bugves of $ 120 $ 120 $ 120。此外，由于系统的抛物线性质，我们研究了正则化效应。这种良好的结果的应用是Mantegazza-Novaga-Tortorelli的“按平面网络的曲率进行运动”中定理3.18的新证明，其中描述了解决方案在存在的最大时期的可能行为。我们的研究是由Mantegazza-Novaga-Pluda-Schulze的“具有多个连接的网络的演变”中提出的一个开放问题的动机：是否存在具有最初基准网络的曲率曲率的独特解决方案？我们给出一个积极的答案。
We prove existence and uniqueness of the motion by curvatureof networks in $\mathbb{R}^n$ when the initial datum is of class $W^{2-\frac{2}{p}}_p$, with triple junction where the unit tangent vectors to the concurring curves form angles of $120$ degrees. Moreover we investigated the regularization effect due to the parabolic nature of the system. An application of this wellposedness result is a new proof of Theorem 3.18 in "Motion by Curvature of Planar Networks" by Mantegazza-Novaga-Tortorelli where the possible behaviors of the solutions at the maximal time of existence are described. Our study is motivated by an open question proposed in "Evolution of Networks with Multiple Junctions " by Mantegazza-Novaga-Pluda-Schulze: does there exist a unique solution of the motion by curvature of networks with initial datum a regular network of class $C^2$? We give a positive answer.
