For an open subset of the Euclidean space of dimension n we consider interior and exterior approximations by sequences of open sets. We prove convergence everywhere of the corresponding sequences of distance functions from boundary as well as convergence almost everywhere for their gradients. As applications we obtain several new Hardy-type inequalities that contain the scalar product of gradients of test functions and the gradient of the distance function from the boundary of an open subset of the Euclidean space.

中文翻译：

---

欧氏空间开放集上距离函数的性质和应用

对于维数为n的欧几里德空间的一个开放子集，我们考虑通过一系列开放集进行内部和外部近似。我们证明了距边界的距离函数的相应序列的各处都存在收敛，并且它们的梯度几乎处处都有收敛。在应用程序中，我们获得了几个新的Hardy型不等式，这些不等式包含测试函数的梯度和距离函数的欧氏空间子集的边界的距离的梯度的标量积。