Monotonicity, continuity and differentiability results for the $L^p$ Hardy constant

TL;DR

This paper investigates how the best constant in the $L^p$ Hardy inequality varies with $p$ for domains in Euclidean space, establishing monotonicity, continuity, and differentiability, especially for non-convex domains where explicit values are unknown.

Contribution

It provides new theoretical results on the dependence of the Hardy constant on $p$, including monotonicity, continuity, and differentiability, for non-convex domains.

Findings

Proves monotonicity of the Hardy constant with respect to $p$

Establishes continuity and differentiability properties of the constant as a function of $p$

Analyzes the case of non-convex domains where explicit constants are not known

Abstract

We consider the $L^p$ Hardy inequality involving the distance to the boundary for a domain in the $n$-dimensional Euclidean space. We study the dependence on $p$ of the corresponding best constant and we prove monotonicity, continuity and differentiability results. The focus is on non-convex domains in which case such constant is in general not explicitly known.