Extremal boundedness of a variational functional in point vortex mean field theory associated with probability measures

TL;DR

This paper investigates the boundedness of a variational functional related to point vortex mean field theory, establishing conditions under which it remains bounded at extremal parameters using advanced estimates.

Contribution

It introduces a new analysis of a Trudinger-Moser type functional associated with probability measures, highlighting the role of residual vanishing in boundedness.

Findings

Boundedness occurs at extremal parameters when residual vanishing happens.

Utilizes a variant of the Y.Y. Li estimate for the analysis.

Provides conditions for the boundedness of the functional.

Abstract

We study a variational functional of Trudinger-Moser type associated with one-sided Borel probability measure. Its boundedness at the extremal parameter holds when the residual vanishing occurs. In the proof we use a variant of the Y.Y. Li estimate.