On the boundary terms in Hardy's inequalities for $W^{1,p}$ functions

TL;DR

This paper derives new Hardy-type inequalities for $W^{1,p}$ functions on bounded star domains using radially invariant vector fields, allowing for explicit maximizers and extending beyond classical boundary term results.

Contribution

It introduces Hardy inequalities for $W^{1,p}$ functions on star domains without boundary terms, which are not derivable from classical $W^{1,p}_0$ inequalities, and explicitly characterizes their maximizers.

Findings

Derived boundary-term-free Hardy inequalities for $W^{1,p}$ functions.

Identified explicit maximizers for these inequalities.

Extended the scope of Hardy inequalities beyond classical $W^{1,p}_0$ cases.

Abstract

With the help of a radially invariant vector field, we derive inequalities of the Hardy kind, with no boundary terms, for $W^{1,p}$ functions on bounded star domains. Our results are not obtainable from the classical inequalities for $W^{1,p}_{0}$ functions. Unlike in $W^{1,p}_{0}$, our inequalities admit maximizers that we describe explicitly.