3:00 p.m., Friday (September 12, 2008)
Department of Mathematics
National Taiwan University
Existence and uniqueness of solution to a mean field equation
in two dimension at the critical parameter
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).