EXTREMAL-FUNCTIONS FOR THE TRUDINGER-MOSER INEQUALITY IN 2 DIMENSIONS

We prove that the Trudinger-Moser constant sup {integral-OMEGA exp (4-pi-u2)dx: u is-an-element-of H-0(1,2)(OMEGA), integral-OMEGA \del u\2 dx less-than-or-equal-to 1} is attained on every 2-dimensional domain. For disks this result is due to Carleson Chang. For other domains we derive 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 Jurgen Moser and Michael Struwe for helpful advice and criticism.