In this paper we relax the current regularity theory for the eikonal equation by using the recent theory of set-valued iterated Lie brackets. We give sufficient conditions for small time local attainability of general, symmetric, nonlinear systems, which have as a consequence the Hölder regularity of the minimum time function in optimal control. We then apply such result to prove Hölder continuity of solutions of the Dirichlet boundary value problem for the eikonal equation with low regularity of the coefficients. We also prove that the sufficient conditions for the Hölder regularity are essentially necessary, at least for smooth vector fields and target.

在本文中，我们通过使用最新的集值迭代李括号理论来放宽当前对电子方程的正则性理论。我们为一般，对称，非线性系统的小时间局部可实现性提供了充分的条件，因此，在最优控制中具有最小时间函数的Hölder规律性。然后，我们将这些结果用于证明具有低系数正则性的方程方程的Dirichlet边值问题的解的Hölder连续性。我们还证明，至少对于光滑的矢量场和目标而言，Hölder正则性的充分条件是必不可少的。