2010/06/14 by Kyril Tintarev, Tintarev, Kyril
Computer Science · Mathematics · #35J20 #35J60 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP #msc:35J20 #msc:35J60
paper · pdf · doi:10.48550/arxiv.1006.2724
openalex publication_date 2010/06/14 · arxiv created 2010/07/16 · arxiv updated 2010/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
While the critical nonlinearity ∫ |u|2^* for the Sobolev space H1 in dimension N>2 lacks weak continuity at any point, Trudinger-Moser nonlinearity ∫ e4πu2 in dimension N=2 is weakly continuous at any point except zero. In the former case the lack of weak continuity can be attributed to invariance with respect to actions of translations and dilations. The Sobolev space H01 of the unit disk \mathbb D⊂\R2 possesses transformations analogous to translations (Möbius transformations) and nonlinear dilations r↦ rs. We present improvements of the Trudinger-Moser inequality with sharper nonlinearities sharper than ∫ e4πu2, that lack weak continuity at any point and possess (separately), translation and dilation invariance. We show, however, that no nonlinearity of the form ∫ F(|x|,u(x))dx is both dilation- and Möbius shift-invariant. The paper also gives a new, very short proof of the conformal-invariant Trudinger-Moser inequality obtained recently by Mancini and Sandeep and of a sharper version of Onofri-type inequality of Beckner.