Let be a compact Riemannian surface with the volume . 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

with

where is a real number.

The mean field equation has a variational structure, and is a solution if and only if it is a critical point of

defined for with

.

It is worth noticing from this entry that so far our Moser-Trudinger’s inequality is just for

**Theorem **(Moser-Trudinger’s inequality for ). There are constants and such that for each

for all .

Thank to a work due to Luigi Fontana [here] we actually have Moser-Trudinger’s inequality for general manifold of a higher order gradient of function. Presicely,

**Theorem **(Moser-Trudinger’s inequality for general manifolds). Let be a compact Riemannian manifold of dimension and a positive integer strictly smaller than . There exists a constant such that for all with

and

the following uniform inequality holds

where the constant is given below

Consequently, for a general Riemannian surface, i.e. , one gets and thus

**Corollary**. There holds

.

For a general function , we have

which after integrating implies

.

Hence, if we deduce

.

Consequently, we get

.

The above inequality is about to say if , the functional is bounded from below, and thus has global minimizer which turns out to be a critical point, hence, a solution. When , the functional is still bounded from below (by a different approach) but it is not when . The case of is not completely solved. We refer the reader to works of Ohtsuka, Malchiodi, Jost, etc.

### Like this:

Like Loading...

*Related*

## Leave a Reply