On the Weighted Hardy Type Inequality for Functions from $W^1_p$ Vanishing on Small Parts of the Boundary

TL;DR

This paper proves a new weighted Hardy-type inequality for Sobolev space functions that vanish on small boundary parts, extending classical inequalities.

Contribution

It introduces a generalized weighted Hardy inequality for functions in Sobolev spaces with boundary vanishing conditions.

Findings

Established a new weighted Hardy inequality for $W^1_p$ functions.

Generalized classical Hardy inequalities to functions vanishing on small boundary parts.

Provides a broader framework for Hardy inequalities in Sobolev spaces.

Abstract

A new weighted Hardy-type inequality for functions from the Sobolev space $W_{p}^{1}$ is proved. It is assumed that functions vanish on small alternating pieces of the boundary. The proved inequality generalizes the classical known weighted Hardy-type inequalities.