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)


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

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)


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