论文标题
局部严格的单数特征ii：$ \ mathbb {r}^2 $的固定方程的存在
Local strict singular characteristics II: existence for stationary equation on $\mathbb{R}^2$
论文作者
论文摘要
严格的奇异特征的概念在粘度解决方案的切割基因座的奇异动态问题中很重要。我们提供了直观且严格的证明，证明了汉密尔顿 - 雅各比方程的严格奇异特性$ h（x，du（x），x），u（x），u（x））= 0 $在二维情况下。我们还证明了如果$ \ mathbf {x} $是一个严格的奇异特征，那么我们确实具有$ \ mathbf {x} $的正确分辨率，而每一个$ t $的$ \ dot {\ mathbf {x}}^+（t）$ \ dot {\ mathbf {x}}^+（t）的直率。如此严格的单数特征必须给出$ p（t）\在d^+u（\ mathbf {x}（t））$中的选择，以使$ p（t）= \ arg arg \ min_ {p \ in d^+u（\ mathbf {x}}（x}（x}（x}（t）（t））} h（x）
The notion of strict singular characteristics is important in the wellposedness issue of singular dynamics on the cut locus of the viscosity solutions. We provide an intuitive and rigorous proof of the existence of the strict singular characteristics of Hamilton-Jacobi equation $H(x,Du(x),u(x))=0$ in two dimensional case. We also proved if $\mathbf{x}$ is a strict singular characteristic, then we really have the right-differentiability of $\mathbf{x}$ and the right-continuity of $\dot{\mathbf{x}}^+(t)$ for every $t$. Such a strict singular characteristic must give a selection $p(t)\in D^+u(\mathbf{x}(t))$ such that $p(t)=\arg\min_{p\in D^+u(\mathbf{x}(t))}H(\mathbf{x}(t),p,u(\mathbf{x}(t)))$.
