New weighted Hardy's inequalities with application to non-existence of global solutions

TL;DR

This paper establishes new weighted Hardy inequalities for a range of p and dimensions, deriving related inequalities and applying them to prove the non-existence of solutions for certain parabolic equations.

Contribution

It introduces novel weighted Hardy inequalities using vector field methods and applies these to demonstrate non-existence results for p-Kolmogorov equations.

Findings

Weighted Hardy inequalities for 1<p<+∞ and d≥1.

Derivation of Poincaré inequalities when p>d.

Non-existence of solutions for specific parabolic equations.

Abstract

In this article, we prove a weighted Hardy inequality for $1<p<+\infty$ and dimension $d\ge1$. If $p>d$, then we can deduce from our weighted Hardy inequality a Poincar\'e inequality. The proof of the weighted Hardy inequality is based on the method of vector fields firstly introduced by Mitidieri \cite{MR1769903}. By the same method, we show for $1<p<+\infty$ and $d\ge1$ that weighted Caffarelli-Kohn-Nirenberg inequalities hold true. As an application of our weighted Hardy inequality, we prove a non-existence result for a p-Kolmogorov parabolic equation.