Hardy type inequalities involving gradient of distance function

Abstract

We prove several new Hardy type inequalities in Euclidean domains; these inequalities involve the gradient of the distance function from a point to the boundary of the domain. For test functions we consider improved inequalities in form proposed by Balinsky and Evans for convex domains. Namely, in Hardy type inequalities, instead of the gradient of the test function, one takes the scalar product of the gradients of the test function and of the distance from a point to the boundary of a given domain. In the present paper, integral Hardy type inequalities are studied in non-convex $n$-dimensional domains having a finite inradius. We prove three new Hardy type $L_p$-inequalities in an improved form with explicit estimates for the constants depending on the dimension of the Euclidean space $n\geq 2$, the inradius of the domain and two parameters $p\geq 1$, $s \geq n$. Our proofs are based on three key ingredients. The first of them is related with an approximation and a special partition of the domain, in particular, we employ the approximation of the domain by subsets formed by finitely many cubes with sides parallel to the coordinate planes. The second ingredient is the representation of the domain as a countable union of subdomains with piece-wise smooth boundaries and applying a new theorem by the author on convergence of the gradients of the distance functions for these subdomains. Moreover, we prove three new Hardy type inequalities on a finite interval, which are employed in justifying the inequalities in multi-dimensional domains.