The Moser-Trudinger-Onofri inequality

Abstract

This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev inequality in higher dimensions. After justifying this statement by recovering the Onofri inequality through various limiting procedures and after reviewing some known results, the authors state several elementary remarks.Various new results are also proved in this paper. A proof of the inequality is given by using mass transportation methods (in the radial case), consistently with similar results for Sobolev inequalities. The authors investigate how duality can be used to improve the Onofri inequality, in connection with the logarithmic Hardy-Littlewood-Sobolev inequality. In the framework of fast diffusion equations, it is established that the inequality is an entropy-entropy production inequality, which provides an integral remainder term. Finally, a proof of the inequality based on rigidity methods is given and a related nonlinear flow is introduced.

A singular Sphere Covering Inequality: uniqueness and symmetry of solutions to singular Liouville-type equations

“…where Ω ⊂ R 2 is a smooth bounded domain and h(x) is a positive function. The latter equation (and its counterpart on manifolds, see (15) below) has been widely discussed in the last decades since it arises in several problems of mathematics and physics, such as Electroweak and Chern-Simons self-dual vortices [65,67,75], conformal geometry on surfaces [71,46,24,25], statistical mechanics of two-dimensional turbulence [20] and of self-gravitating systems [74] and cosmic strings [61], theory of hyperelliptic curves [22], Painlevé equations [27] and Moser-Trudinger inequalities [18,37,40,45,60]. There are by now many results concerning existence and multiplicity [3,7,8,9,11,15,16,21,30,32,34,35,36,54,57,58], uniqueness [12,13,14,42,44,50,52,66], blow-up phenomena [6,10,17,…”

Onofri inequalities and rigidity results

“…One has to mention that the sphere M = S 2 is an important case of application of our method, for which other types of remainder terms can be produced. See [24] for more details. It has to be noted that on S 2 we have λ ⋆ = ρ = λ 1 /2 = 1.…”

“…We will give a negative answer in Section 1. Corollary 2 is established in Section 2 using a nonlinear flow that has already been considered on the sphere in [24]. The case d = 1 is very simple and will be considered for the sake of completeness in Section 3.…”

A dual Moser–Onofri inequality and its extensions to higher dimensional spheres

“…We then deduce the following higher dimensional version of the Euclidean Onofri inequality in R n , n ≥ 2: (1.12) and u ≡ 0 is the extremal. We finish this introduction by mentioning that the Euclidean Onofri inequality (1.5) in the radial case was recently obtained in [6] using mass transport techniques. However, to our knowledge, our duality result, the extensions of Onofri's inequality to higher dimensions, as well as the mass transport proof of the general (non-radial) Onofri inequality are new.…”