# Ngô Quốc Anh

## January 23, 2011

### Equivalent forms of the mean field equations

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 1:36

Let $(M,g)$ be a compact Riemannian surface, $h$ be a positive $C^1$ function on $M$. The standard mean field equation can be stated as follows

$\displaystyle {\Delta _g}w + \rho \left( {\frac{{h(x){e^w}}}{{\displaystyle\int_M {h(x){e^w}} }} - 1} \right) = 4\pi \sum\limits_{j = 1}^m {{\alpha _j}({\delta _{{p_j}}} - 1)}$

in $M$, where $p_j \in M$ are given distinct points, $\alpha_j >0$ and $\delta_j$ denotes the Dirac measure with pole at $p_j$. Here the area of $M$ is assumed to be constant $1$ and $\Delta_g$ stands for the Laplace Beltrami operator with respect to $g$.

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

$\displaystyle \int_M w=0$.

Let $G(x,p)$ be the Green function with pole at $p$, that is,

$\displaystyle\begin{cases}-\Delta_g G(x,p)=\delta_p-1,&{\rm in}\; M,\\\displaystyle\int_M G(x,p)=0,\end{cases}$

and let

$\displaystyle u(x) = w(x) + 4\pi \sum\limits_{j = 1}^m {{\alpha _j}G(x,{p_j})}$.

In terms of $u$, one has

$\displaystyle\begin{gathered} {\Delta _g}u = {\Delta _g}w + {\Delta _g}\left( {4\pi \sum\limits_{j = 1}^m {{\alpha _j}G(x,{p_j})} } \right) \hfill \\ \qquad= - \rho \left( {\frac{{h(x){e^w}}}{{\displaystyle\int_M {h(x){e^w}} }} - 1} \right) + 4\pi \left[ {\sum\limits_{j = 1}^m {{\alpha _j}({\delta _{{p_j}}} - 1)} + \sum\limits_{j = 1}^m {{\alpha _j}{\Delta _g}G(x,{p_j})} } \right] \hfill \\ \qquad= - \rho \left( {\frac{{h(x)\exp ( - 4\pi \sum\limits_{j = 1}^m {{\alpha _j}G(x,{p_j})} ){e^u}}}{{\displaystyle\int_M {h(x)\exp ( - 4\pi \sum\limits_{j = 1}^m {{\alpha _j}G(x,{p_j})} ){e^u}} }} - 1} \right) \hfill \\ \qquad= - \rho \left( {\frac{{W(x){e^u}}}{{\displaystyle\int_M {W(x){e^u}} }} - 1} \right) \hfill \\ \end{gathered}$

where

$\displaystyle W(x) = h(x)\prod\limits_{j = 1}^m {\exp ( - 4\pi {\alpha _j}G(x,{p_j}))}$.

Thus, $W(x) \in C^0(M)$ and vanishes exactly at $\{p_1,...,p_m\}$. The mean field equation now reads as follows

$\displaystyle {\Delta _g}u + \rho \left( {\frac{{W(x){e^u}}}{{\displaystyle\int_M {W(x){e^u}} }} - 1} \right) = 0$.

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

1. 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

2. 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