$L^p$ Hardy inequality on $C^{1,\gamma}$ domains

TL;DR

This paper extends the understanding of the $L^p$ Hardy inequality to $C^{1,eta}$ domains, providing new existence, non-existence, and decay estimates for minimizers in both bounded and exterior domains.

Contribution

It generalizes known results from $C^2$ to $C^{1,eta}$ domains, including new estimates and non-existence results for exterior domains.

Findings

Extended Hardy inequality results to $C^{1,eta}$ domains.

Established decay estimates for minimizers in exterior domains.

Proved new non-existence results for certain domain classes.

Abstract

We consider the $L^p$ Hardy inequality involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty compact boundary. We extend the validity of known existence and non-existence results, as well as the appropriate tight decay estimates for the corresponding minimizers, from the case of domains of class $C^2$ to the case of domains of class $C^{1,\gamma}$ with $\gamma \in (0,1]$. We consider both bounded and exterior domains. The upper and lower estimates for the minimizers in the case of exterior domains and the corresponding related non-existence result seem to be new even for $C^2$-domains.