# Ngô Quốc Anh

## July 1, 2010

### The Trudinger inequality

Filed under: PDEs — Tags: , — Ngô Quốc Anh @ 10:53

In 1967, Neil S. Trudinger announced a result in J. Math. Mech. (now known as Indiana Univ. Math. J.) which can be seen as a limiting case of the Sobolev inequality [here] or [here].

It is well-known from the Sobolev embedding theorem that

$W_0^{\alpha, q}(\Omega) \hookrightarrow L^p(\Omega)$

for

$\displaystyle \frac{1}{p}=\frac{1}{q}-\frac{\alpha}{n}, \quad q\alpha.

The case $q\alpha=n$ is commonly referred to the limitting case. If $\alpha=1$, $n=2$ and $q<2$ we obtain

$W_0^{1, q}(\Omega) \hookrightarrow L^p(\Omega)$.

In general one cannot take the limits $q \to 2$ and $p \to \infty$, i.e.

$W_0^{1, 2}(\Omega) \not\hookrightarrow L^\infty(\Omega)$.

A counter-example is given by

$\displaystyle \log\left(1+\log\frac{1}{|x|}\right)$