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).

(more…)

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