The Moser-Trudinger-Onofri inequality
Dolbeault, Jean; Esteban, Maria J.; Jankowiak, Gaspard (2015), The Moser-Trudinger-Onofri inequality, Chinese Annals of Mathematics. Series B, 36, 5, p. 777-802. 10.1007/s11401-015-0976-7
TypeArticle accepté pour publication ou publié
External document linkhttps://arxiv.org/abs/1403.5042v3
Journal nameChinese Annals of Mathematics. Series B
Shanghai Scientific and Technological Literature Publishing House
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Abstract (EN)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.
Subjects / KeywordsMoser-Trudinger-Onofri inequality; Duality; Mass transportation; Fast diffusion equation; Rigidity
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