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})}.

In terms of u, one has

\displaystyle\begin{gathered} {\Delta _g}u = {\Delta _g}w + {\Delta _g}\left( {4\pi \sum\limits_{j = 1}^m {{\alpha _j}G(x,{p_j})} } \right) \hfill \\ \qquad= - \rho \left( {\frac{{h(x){e^w}}}{{\displaystyle\int_M {h(x){e^w}} }} - 1} \right) + 4\pi \left[ {\sum\limits_{j = 1}^m {{\alpha _j}({\delta _{{p_j}}} - 1)} + \sum\limits_{j = 1}^m {{\alpha _j}{\Delta _g}G(x,{p_j})} } \right] \hfill \\ \qquad= - \rho \left( {\frac{{h(x)\exp ( - 4\pi \sum\limits_{j = 1}^m {{\alpha _j}G(x,{p_j})} ){e^u}}}{{\displaystyle\int_M {h(x)\exp ( - 4\pi \sum\limits_{j = 1}^m {{\alpha _j}G(x,{p_j})} ){e^u}} }} - 1} \right) \hfill \\ \qquad= - \rho \left( {\frac{{W(x){e^u}}}{{\displaystyle\int_M {W(x){e^u}} }} - 1} \right) \hfill \\ \end{gathered}

where

\displaystyle W(x) = h(x)\prod\limits_{j = 1}^m {\exp ( - 4\pi {\alpha _j}G(x,{p_j}))} .

Thus, W(x) \in C^0(M) and vanishes exactly at \{p_1,...,p_m\}. The mean field equation now reads as follows

\displaystyle {\Delta _g}u + \rho \left( {\frac{{W(x){e^u}}}{{\displaystyle\int_M {W(x){e^u}} }} - 1} \right) = 0.

We will discuss some techniques related to such an equation later.

4 Comments »

  1. well done!
    Have you got “AN EXPOSITORY SURVEY ON THE RECENT DEVELOPMENT OF MEAN FIELD EQUATIONS” by Chang-Shou Lin, please?
    Thanks

    Comment by Fab — January 25, 2011 @ 17:45

    • Unfortunately I have no copy of that article but you may access its electronic version through the following DOI at http://dx.doi.org/10.3934/dcds.2007.19.387. Amazingly you are also interested in the mean feld equations. Btw, I strongly recommend you to go through the following book entitled “Selfdual Gauge Field Vortices – An Analytical Approach” by Gabriella Tarantello.

      Comment by Ngô Quốc Anh — January 25, 2011 @ 19:48

  2. I know this amazing book, I’m studying it right now!
    I’m a young PHD student from Italy, from Rome so I know her pretty good.
    Thanks so much for the many interesting things in your website!

    Comment by Fab — January 25, 2011 @ 21:29

    • So am I but from the National University of Singapore.

      Comment by Ngô Quốc Anh — January 25, 2011 @ 22:45


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