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Special Session 73: Nonlinear elliptic and parabolic equations and related functional inequalities

N-Euclidean Logarithmic Moser-Trudinger-Onofri inequality and some geometrical variants

Gabriele Mancini

University of Bari Aldo Moro
Italy

Co-Author(s): Natalino Borgia, Silvia Cingolani

Abstract:

In this talk, I will provide a brief overview of the history of Onofri`s inequality for the unit sphere, highlighting its connection with Trudinger-Moser type inequalities on Euclidean bounded domains. I will focus on an

$N$ -dimensional Euclidean version of Onofri`s inequality, proved by del Pino and Dolbeault, for smooth compactly supported functions in

$\mathbb{R}^N$ , with

$N \geq 2$ . I will prove that this inequality can be extended to a suitable weighted Sobolev space and, although there is no clear connection with standard Sobolev spaces on

$\mathbb{S}^N$ via stereographic projection, I will show that it is equivalent to the logarithmic Moser-Trudinger inequality with sharp constant, obtained by Carleson and Chang for balls in

$\mathbb{R}^N$ . These results are part of a joint work with N. Borgia and S. Cingolani.

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