论文标题
$ \ mathbb {r}^2 $上的本地单数特征
Local singular characteristics on $\mathbb{R}^2$
论文作者
论文摘要
众所周知，汉密尔顿 - 雅各比方程的奇异粘度解决方案是从任何非临界奇异点传播的，沿奇异特征沿着满足某些差异夹杂物的曲线。在文献中，引入了不同的奇异特征概念。然而，目前缺少了奇异特性的一般唯一性标准，而不仅限于一个空间维度的机械系统或问题。在本文中，我们证明，对于$ \ mathbb {r}^2 $上的Tonelli Hamiltonian来说，对单数特征的两个不同的概念恰好是Bi-lipschitz重新集化。重要的是，我们获得了Khanin和Sobolevski在论文中引入的单数特征类别的唯一性结果[汉密尔顿 - 雅各比方程的拉格朗日轨迹动力学。拱。配给。机械。肛门，219（2）：861-885，2016]。
The singular set of a viscosity solution to a Hamilton-Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the literature, different notions of singular characteristics were introduced. However, a general uniqueness criterion for singular characteristics, not restricted to mechanical systems or problems in one space dimension, is missing at the moment. In this paper, we prove that, for a Tonelli Hamiltonian on $\mathbb{R}^2$, two different notions of singular characteristics coincide up to a bi-Lipschitz reparameterization. As a significant consequence, we obtain a uniqueness result for the class of singular characteristics that was introduced by Khanin and Sobolevski in the paper [On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations. Arch. Ration. Mech. Anal., 219(2):861-885, 2016].
