# 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

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

## August 13, 2010

### An application of the (Moser-)Trudinger inequality to the mean field equations

Let $(M,g)$ be a compact Riemannian surface with the volume $|M|$. The simplest form of the mean field equation studied in the contexts of the prescribing Gaussian curvature, statistical mechanics of many vortex points in the perfect fluid and self-dual gauss theories is given by

$\displaystyle - {\Delta _g}u = \lambda \left( {\frac{{{e^u}}}{{\int_M {{e^u}d{v_g}} }} - \frac{1}{{|M|}}} \right), \quad \text{ on } M$

with

$\displaystyle\int_M {ud{v_g}} = 0$

where $\lambda$ is a real number.

The mean field equation has a variational structure, and $u$ is a solution if and only if it is a critical point of

$\displaystyle {J_\lambda }(v) = \frac{1}{2}\int_M {{{\left| {\nabla v} \right|}^2}d{v_g}} - \lambda \log \int_M {{e^v}d{v_g}}$

defined for $v \in H^1(M)$ with

$\displaystyle\int_M {vd{v_g}} = 0$.

It is worth noticing from this entry that so far our Moser-Trudinger’s inequality is just for $\mathbb S^2$

Theorem (Moser-Trudinger’s inequality for $\mathbb S^2$). There are constants $\eta>0$ and $c=c(g)>0$ such that for each $p \geqslant 2$

$\displaystyle \log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \left[ {\frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u } \right] + {C_2}$

for all $u \in W^{1,2}(\mathbb S^2)$.