正则网络曲率运动的存在唯一性

IF 1.2 4区 数学 Q1 MATHEMATICS
Michael Gosswein, Julia Menzel, Alessandra Pluda
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引用次数: 10

摘要

我们证明了在$\mathbb{R}^n$中,当初始基准为$W^{2} - $ frac{2}{p}}_p$时,网络曲率运动的存在性和唯一性,其中共曲线的单位切向量形成$120°角。此外,我们还研究了由于系统的抛物性质而引起的正则化效应。这个适位性结果的一个应用是Mantegazza-Novaga-Tortorelli在“平面网络的曲率运动”中对定理3.18的一个新的证明,其中描述了解在最大存在时间的可能行为。我们的研究是由Mantegazza-Novaga-Pluda-Schulze在“具有多结点的网络的进化”中提出的一个开放问题所激发的:是否存在一个具有初始基准的网络的曲率运动的唯一解,一个类为C^2的规则网络?我们给出一个肯定的答案。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Existence and uniqueness of the motion by curvature of regular networks
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.
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来源期刊
CiteScore
1.70
自引率
0.00%
发文量
17
审稿时长
>12 weeks
期刊介绍: Interfaces and Free Boundaries is dedicated to the mathematical modelling, analysis and computation of interfaces and free boundary problems in all areas where such phenomena are pertinent. The journal aims to be a forum where mathematical analysis, partial differential equations, modelling, scientific computing and the various applications which involve mathematical modelling meet. Submissions should, ideally, emphasize the combination of theory and application.
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