Abstract

We prove new integral inequalities for real-valued test functions defined on subdomains of the Euclidean space. Namely, we obtain several new Hardy-type inequalities that contain the scalar product of gradients of test functions and of the gradient of the distance function from the boundary of an open subset of the Euclidean space. Our method of proof is based on interior and exterior approximations of a given domain by sequences of simplest domains and has two important ingredients. The first one is approximations of a given domain by elementary domains that admit a special partition. The second ingredient is presented in this paper by Theorems 1 and 2 about convergence everywhere of the sequences of distance functions from boundary of approximating elementary domains as well as about convergence almost everywhere for their gradients. In the proofs we also use some basic theorems due to Rademacher, Hardy, Motzkin and Hadwiger.

中文翻译：

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欧氏空间域上的Hardy型不等式的强形式

摘要

我们证明了在欧几里得空间的子域上定义的实值测试函数的新积分不等式。即，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的开放子集的边界的距离的梯度的标量积。我们的证明方法基于给定域的最简单域序列的内部和外部近似，并且具有两个重要成分。第一个是给定域的近似值，其中包含允许特殊分区的基本域。定理1和2在本文中提出了第二个成分，关于距离函数序列到近似基本域边界的收敛性以及关于梯度的几乎所有收敛性。