Trudinger's theorem

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In mathematical analysis, Trudinger's theorem or the Trudinger inequality (sometimes called Moser-Trudinger inequality) is a result of functional analysis on Sobolev spaces.

It provides an inequality between a certain Sobolev space norm and an Orlicz space norm of a function, named for Neil Trudinger (and Jürgen Moser). The inequality is a limiting case of Sobolev imbedding and can be stated as the following theorem:

Let Ω be a bounded domain in \mathbb{R}^n satisfying the cone condition. Let mp = n and p > 1. Set

A(t)=\exp\left( t^{n/(n-m)} \right)-1.

Then there exists the imbedding

W^{m,p}(\Omega)\hookrightarrow L_A(\Omega)

where

L_A(\Omega)=\left\{ u\in M_f(\Omega):\|u\|_{A,\Omega}=\inf\{ k>0:\int_\Omega A\left( \frac{|u(x)|}{k} \right)~dx\leq 1 \} \right\}.

The space

LA(Ω)

is an example of an Orlicz space.

[edit] See also

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