Functional aspects of the Hardy inequality. Appearance of a hidden energy

TL;DR

This paper explores a reformulation of the Hardy inequality and associated evolution problems, revealing a hidden energy component that affects the total energy in bounded and unbounded domains, including effects from infinity.

Contribution

It introduces a new perspective on the Hardy inequality by identifying a hidden energy term and analyzing its impact on the energy functional in various domain settings.

Findings

Discovery of a nontrivial Hardy singularity energy affecting total energy

Identification of a hidden energy component in the Hardy inequality

Revelation that the hidden energy can dominate the total energy in exterior domains

Abstract

Starting with a functional difficulty appeared in the paper \cite{vz00} by V\'azquez and Zuazua, we obtain new insights into the Hardy Inequality and the evolution problem associated to it by means of a reformulation of the problem. Surprisingly, the connection of the energy of the new formulation with the standard Hardy functional is nontrivial, due to the presence of a Hardy singularity energy. This corresponds to a loss for the total energy. The problem arises when the equation is posed in a bounded domain, and also when posed in the whole space. We also consider an equivalent problem with inverse square potential on an exterior domain. The extra energy term is then present as an effect that comes from infinity, a kind of hidden energy. In this case, in an unexpected way, this term is additive to the total energy, and it may even constitute the main part of it.