Existence and uniqueness of solution to a mean field equation in two dimension at the critical parameter

Abstract: The existence of solutions to

\Delta u + 8\pi e^u / (\int e^u) = 0 in \Omega
u = 0 on \partial \Omega

where \Omega is a bounded domain of \R^2 depends on the geometry of \Omega . For example, if \Omega is a ball, then the equation has no solutions. But, for a long and thin ellipse, solutions exist. In this talk, I will give a sufficient and necessary condition for the existence of solutions. This condition is expressed in terms of the regular part of the Green function of \Omega .

I will also talk about the uniqueness problem for this equation.

Refreshments will be served at 2:45 p.m. (Math Lounge, MATX 1115).