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arXiv 1702.07618math.OCmath.AP
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Hörmander向量场最小时间函数的正则性结果

Regularity results for the minimum time function with Hörmander vector fields

Paolo Albano, Piermarco Cannarsa, Teresa Scarinci

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摘要：

在具有光滑边界的有界$\mathbb{R}^n$区域上，研究了一类光滑向量场$\{X_1,\ldots ,X_N\}$所对应的程函方程狄利克雷问题在Hörmander括号生成条件下黏性解$T$的正则性.由于特征边界点的存在，在这种情况下可能出现奇异轨迹。我们的特点是这样的轨迹的所有点的闭集的解决方案失去逐点Lipschitz连续性。然后，我们证明了局部Lipschitz连续性的$T$，当地的可压缩腔的$T$，和没有奇异轨迹是等价的性质。最后，我们证明了当$\{X_1,\ldots ,X_N\}$的特征集是一个辛流形时，满足最后一个条件。我们应用我们的结果海森堡和Martinet的向量场。

英文摘要：

In a bounded domain of $\mathbb{R}^n$ with smooth boundary, we study the regularity of the viscosity solution, $T$, of the Dirichlet problem for the eikonal equation associated with a family of smooth vector fields $\{X_1,\ldots ,X_N\}$, subject to Hörmander's bracket generating condition. Due to the presence of characteristic boundary points, singular trajectories may occur in this case. We characterize such trajectories as the closed set of all points at which the solution loses point-wise Lipschitz continuity. We then prove that the local Lipschitz continuity of $T$, the local semiconcavity of $T$, and the absence of singular trajectories are equivalent properties. Finally, we show that the last condition is satisfied when the characteristic set of $\{X_1,\ldots ,X_N\}$ is a symplectic manifold. We apply our results to Heisenberg's and Martinet's vector fields.

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