# Ngô Quốc Anh

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