[Paper Review] The Moser-Trudinger-Onofri inequality

Jean Dolbeault, Maria J. Esteban+1 more|Mar 20, 2014

Nonlinear Partial Differential Equations27 references4 citations

TL;DR

This paper provides a comprehensive analysis of the Moser-Trudinger-Onofri inequality in two dimensions, establishing it as a sharp functional inequality equivalent to Sobolev inequalities in higher dimensions. It introduces novel proofs using mass transportation, duality with the logarithmic Hardy-Littlewood-Sobolev inequality, and a nonlinear flow based on the carré du champ method, yielding an integral remainder term and proving optimality through rigidity and entropy-entropy production arguments on the sphere and related spaces.

ABSTRACT

This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or Onofri inequality for brevity. In dimension two this inequality plays a role similar to 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, we state several elementary remarks. We also prove various new results. We give a proof of the inequality using mass transportation methods (in the radial case), consistently with similar results for Sobolev's inequalities. We 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, we establish that the inequality is an entropy--entropy production inequality, which provides an integral remainder term. Finally we give a proof of the inequality based on rigidity methods and introduce a related nonlinear flow.

Motivation & Objective

• To establish the Onofri inequality as a sharp analog of Sobolev inequalities in two dimensions through multiple limiting and transformation procedures.
• To provide a new proof using mass transportation methods in the radial case, consistent with known techniques for Sobolev inequalities.
• To explore duality between the Onofri inequality and the logarithmic Hardy-Littlewood-Sobolev inequality to improve the inequality's sharpness.
• To interpret the Onofri inequality as an entropy–entropy production inequality in the context of fast diffusion equations, deriving an integral remainder term.
• To introduce a nonlinear flow based on the carré du champ method that characterizes extremal functions and proves the inequality via convergence to equilibrium.

Proposed method

• Use of stereographic projection and Emden-Fowler transformation to establish equivalence between the Onofri inequality on the Euclidean plane, the sphere, and the cylinder.
• Application of mass transportation techniques in the radial case to derive the inequality, extending methods used for Sobolev inequalities.
• Employment of duality via the logarithmic Hardy-Littlewood-Sobolev inequality to refine and improve the Onofri inequality.
• Formulation of a fast diffusion equation on the sphere, showing that the Onofri inequality arises as an entropy–entropy production inequality with a remainder term.
• Introduction of a nonlinear evolution equation on the sphere: ∂f/∂t = Δ(e^{-f/2}) - (1/2)|∇f|² e^{-f/2}, which drives functions toward equilibrium.
• Use of the carré du champ method to define a non-negative quadratic form Rλ[f] that quantifies the decay of the functional Gλ, leading to an integral remainder in the inequality.

Experimental results

Research questions

• RQ1 How can the Moser-Trudinger-Onofri inequality be derived via mass transportation in the radial case?
• RQ2 In what way does duality with the logarithmic Hardy-Littlewood-Sobolev inequality improve the Onofri inequality?
• RQ3 Can the Onofri inequality be interpreted as an entropy–entropy production inequality in the framework of fast diffusion equations?
• RQ4 What role does the nonlinear flow defined by the carré du champ method play in proving the inequality and characterizing extremal functions?
• RQ5 How do symmetrization and rigidity methods contribute to proving uniqueness and sharpness of the inequality?

Key findings

• The Onofri inequality is proven via mass transportation in the radial case, establishing a direct link to known methods for Sobolev inequalities.
• Duality with the logarithmic Hardy-Littlewood-Sobolev inequality provides a mechanism to improve the Onofri inequality, particularly in the context of sharp constants.
• The inequality is shown to be equivalent to an entropy–entropy production inequality for the fast diffusion equation on the sphere, with an explicit integral remainder term.
• A nonlinear flow on the sphere, governed by the equation ∂f/∂t = Δ(e^{-f/2}) - (1/2)|∇f|² e^{-f/2}, drives initial data toward equilibrium, and the decay of the functional Gλ is quantified by the non-negative functional Rλ[f].
• The carré du champ method yields a non-negative remainder term Rλ[f], and the inequality is proven via integration of dGλ/dt = -Rλ[f] over time, leading to Gλ[v] ≥ ∫₀^∞ Rλ[f(t)] dt.
• For any λ ∈ (0,1], the inequality holds with equality in the limit as t → ∞, and the result extends to general L¹ ∩ H¹ functions by density and lower semicontinuity.