存在障碍物时的距离函数
The distance function in the presence of an obstacle
摘要：
研究了在光滑超曲面包围的紧致障碍物存在的情况下，从欧几里得空间的不动点（逐点目标）到黎曼距离函数的问题。首先，我们表明，这样的功能是局部的半阶的分数模，并且，在障碍物附近，这种规律性是最佳的。然后，在欧几里德设置，我们证明了距离函数的奇点传播，在这个意义上，每个奇点属于一个非平凡的奇异连续。最后，我们研究了凸障碍物存在时距离函数的不可微性。
英文摘要：
We study the Riemannian distance function from a fixed point (a point-wise target) of Euclidean space in the presence of a compact obstacle bounded by a smooth hypersurface. First, we show that such a function is locally semiconcave with a fractional modulus of order one half and that, near the obstacle, this regularity is optimal. Then, in the Euclidean setting, we prove that the singularities of the distance function propagate, in the sense that each singular point belongs to a nontrivial singular continuum. Finally, we investigate the lack of differentiability of the distance function when a convex obstacle is present.
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