Let be a compact Riemannian surface, be a positive function on . The standard mean field equation can be stated as follows

in , where are given distinct points, and denotes the Dirac measure with pole at . Here the area of is assumed to be constant and stands for the Laplace Beltrami operator with respect to .

Clearly, the above PDE is invariant under adding a constant. Hence, is normalized to satisfy

.

Let be the Green function with pole at , that is,

and let

.

In terms of , one has

where

.

Thus, and vanishes exactly at . The mean field equation now reads as follows

.

We will discuss some techniques related to such an equation later.

well done!

Have you got “AN EXPOSITORY SURVEY ON THE RECENT DEVELOPMENT OF MEAN FIELD EQUATIONS” by Chang-Shou Lin, please?

Thanks

Comment by Fab — January 25, 2011 @ 17:45

Unfortunately I have no copy of that article but you may access its electronic version through the following DOI at http://dx.doi.org/10.3934/dcds.2007.19.387. Amazingly you are also interested in the mean feld equations. Btw, I strongly recommend you to go through the following book entitled “Selfdual Gauge Field Vortices – An Analytical Approach” by Gabriella Tarantello.

Comment by Ngô Quốc Anh — January 25, 2011 @ 19:48

I know this amazing book, I’m studying it right now!

I’m a young PHD student from Italy, from Rome so I know her pretty good.

Thanks so much for the many interesting things in your website!

Comment by Fab — January 25, 2011 @ 21:29

So am I but from the National University of Singapore.

Comment by Ngô Quốc Anh — January 25, 2011 @ 22:45