关于Hamilton-Jacobi方程解的正则性和全局结构的注记

Remarks on the Regularity and Global Structure of Solutions to Hamilton-Jacobi Equations

该文考虑高维Hamilton-Jacobi方程的柯西问题.作者证明了从任一初始点出发的特征线永不碰到奇异点集合的充分必要条件是初始函数在该点取到最小值.在此基础上,证明了奇异点集合的道路连通分支和初始函数不取最小值的点集合的道路连通分支之间存在一一对应,而且解的梯度的间断一旦产生就不会消失.特别指出,该文的结果不需要"初始函数梯度在无穷远趋近于零"这一限制条件,而文献[12]中重要的命题2.7和主要结果之一的定理3.3是在这一条件下得到的.

The paper is concerned with the Cauchy problem for the Hamilton-Jacobi equations of multi-dimensional space variables. We prove the sufficient and necessary condition for that a characteristic emanating from a given point never touches the set of singularity points is that the initial function attains its minimum at this point. Finally, we prove there exists one-to-one correspondence between the path connected components of the set of singularity points and the path connected components of the set on which the initial function ctoes not attain its minimum. Each path connected component of the set of the singularity points never terminates at a finite time. In particularly, it is worth pointing out that our results are obtained without assuming that the gradient of the initial function tends to zero at infinity under which the important proposition 2.7 and theorem 3.3, one of the main results in [12] are obtained.