Is the Trudinger-Moser nonlinearity a true critical nonlinearity?

Details Is the Trudinger-Moser nonlinearity a true critical nonlinearity? Is the Trudinger-Moser nonlinearity a true critical nonlinearity?

2010-06-14

arxiv.org/abs/1006.2724v2

Mathematics

Analysis of PDEs

Scientific paper

While the critical nonlinearity $\int |u|^{2^*}$ for the Sobolev space $H^1$ in dimension $N>2$ lacks weak continuity at any point, Trudinger-Moser nonlinearity $\int e^{4\pi u^2}$ 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 $H_0^1$ of the unit disk $\mathbb D\subset\R^2$ possesses transformations analogous to translations (M\"obius transformations) and nonlinear dilations $r\mapsto r^s$. We present improvements of the Trudinger-Moser inequality with sharper nonlinearities sharper than $\int e^{4\pi u^2}$, that lack weak continuity at any point and possess (separately), translation and dilation invariance. We show, however, that no nonlinearity of the form $\int F(|x|,u(x))\mathrm{d}x$ is both dilation- and M\"obius 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.

Affiliated with

Also associated with

No associations

Rating

Is the Trudinger-Moser nonlinearity a true critical nonlinearity? does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Is the Trudinger-Moser nonlinearity a true critical nonlinearity?, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Is the Trudinger-Moser nonlinearity a true critical nonlinearity? will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-422629