# 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

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