Extremal functions for the trudinger-moser inequality in 2 dimensions

We prove that theTrudinger-Moser constant $$\sup \left\{ {\int_\Omega {\exp (4\pi u^2 )dx:u \in H_0^{1,2} (\Omega )\int_\Omega {\left| {\nabla u} \right|^2 dx \leqslant 1} } } \right\}$$ is attained on every 2-dimensional domain. For disks this result, is due to Carleson-Chang. For other domains we derived an isoperimetric inequality which relates the ratio of the supremum of, the functional and its maximal limit on concentrating sequences to the corresponding quantity for disks. A conformal rearrangement is introduced to prove this inequality. I would like to thank Jrgen Moser and Michael Struwe for helpful advice and criticism.