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})}.

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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).

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