Title: Boundary regularity for the distance functions, and the eikonal equation Show Chinese title

Title: 距离函数的边界正则性与几何光学方程

Abstract: We study the gain in regularity of the distance to the boundary of a domain in $\mathbb R^m$. In particular, we show that if the signed distance function happens to be merely differentiable in a neighborhood of a boundary point, it and the boundary have to be $\mathcal C^{1,1}$ regular. Conversely, we study the regularity of the distance function under regularity hypotheses of the boundary. Along the way, we point out that any solution to the eikonal equation, differentiable everywhere in a domain of the Euclidean space, admits a gradient which is locally Lipschitz. Show Chinese abstract

Abstract: 我们研究了域边界距离在$\mathbb R^m$中的规则性增益。 特别是，我们证明了如果符号距离函数在边界点的一个邻域内仅仅是可微的，那么它和边界必须是$\mathcal C^{1,1}$规则的。 反之，我们研究了在边界规则性假设下距离函数的规则性。 在此过程中，我们指出欧几里得空间中域上处处可微的 eikonal 方程的任何解都具有局部Lipschitz梯度。