On an Inequality by N. Trudinger and J. Moser and Related Elliptic Equations

2022.12.21（星期三）19:00-20:00

点：腾讯会议: 685-289-166

要：It has been shown by Trudinger and Moser that for normalized functions $u$ of the Sobolev space $W^{1,N}(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^N$, one has $\int_\Omega exp(\alpha_N |u|^N/(N-1)) dx \leq C_N$, where $\alpha_N$ is an explicit constant depending only on $N$, and $C_N$ is a constant depending only on $N$ and $\Omega$. Carleson and Chang proved that there exists a corresponding extremal function in the case that $\Omega$ is the unit ball in $\mathbb{R}^N$. In this paper, we give a new proof, a generalization, and a new interpretation of this result. As an application, the existence of a nontrivial solution for a related elliptic equation with “Trudinger-Moser” growth is proved. This talk is based on the paper by Bernhard Ruf et al. on CPAM(2002).