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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