Followed by an entry where the Trudinger inequality had been discussed we now consider an important variant of it known as the Moser-Trudinger inequality.
Let us remind the Trudinger inequality
Theorem (Trudinger). Let
be a bounded domain and
with
.
Then there exist universal constants
,
such that
.
The Trudinger inequality has lots of application. For application to the prescribed Gauss curvature equation, one requires a particular value for the best constant . In connection with his work on the Gauss curvature equation, J. Moser [here] sharpended the above result of Trungdier as follows
Theorem (Moser). Let
be a bounded domain and
with
.
Then there exist sharp constants
,
given by
such that
.
The constant
is sharp in the sense that for all
there is a sequence of functions
satisfying
but the integral
grow without bound.
For general compact closed manifold the constant on the right hand side of the Moser-Trudinger inequality depends on the metric
. Working on a sphere
with a canonical metric allows us to control the constants.
Theorem (Moser). There is a universal constant
such that for all
with
and
we have
.
Observe that
.
In the same way as we introduce in the entry concerning the Trudinger inequality one can show
Corollary. For
one has
for all
.
Obviously, since
. It turns out to determine the best constant
. This had been done by Onofri known as the Onofri inequality [here].
Theorem (Onofri).Let
then we have
with the equality iff
.
The proof of the Onofri inequality relies on a result due to Aubin
Theorem (Aubin). For all
there exists a constant
such that
for any
belonging to the following class
.
Source: S-Y.A. Chang, Non-linear elliptic equations in conformal geometry, EMS, 2004.