New analytical and geometrical aspects on Trudinger-Moser type inequality in 2D

TL;DR

This survey reviews the history, geometric connections, and recent advances in Trudinger-Moser inequalities in 2D, highlighting their relation to Onofri's inequality and nonlocal energy functionals with logarithmic kernels.

Contribution

It provides a comprehensive overview of the development and recent progress in Trudinger-Moser inequalities, including new insights into nonlocal energy functionals.

Findings

Connection between Onofri's inequality and Euclidean sharp inequalities

Recent results on nonlocal interaction energy functionals

Historical overview of Trudinger-Moser inequalities in 2D

Abstract

The present survey is devoted to results on Trudinger-Moser inequalities in two dimension. We give a brief overview of the history of these celebrated inequalities and, starting from the geometric problem that motivated Moser's original work, we discuss the connection between Onofri's inequality for the unit sphere and sharp inequalities on Euclidean domains. Finally, we present recent results and new insights into nonlocal interaction energy functionals in two dimension, involving logarithmic kernels.