English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Starting with a functional difficulty appeared in the paper \cite{vz00} byV\'azquez and Zuazua, we obtain new insights into the Hardy Inequality and theevolution problem associated to it by means of a reformulation of the problem.Surprisingly, the connection of the energy of the new formulation with thestandard Hardy functional is nontrivial, due to the presence of a Hardysingularity energy. This corresponds to a loss for the total energy. Theproblem arises when the equation is posed in a bounded domain, and also whenposed in the whole space.  We also consider an equivalent problem with inverse square potential on anexterior domain. The extra energy term is then present as an effect that comesfrom 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 mainpart of it.

Functional aspects of the Hardy inequality: appearance of a hidden energy