Trudinger's theorem

In mathematical analysis, Trudinger's theorem or the Trudinger inequality (also sometimes called the Moser–Trudinger inequality) is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger (and Jürgen Moser).

It provides an inequality between a certain Sobolev space norm and an Orlicz space norm of a function. The inequality is a limiting case of Sobolev imbedding and can be stated as the following theorem:

Let $\mathrm{\Omega }$ be a bounded domain in ${ℝ}^n$ 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 embedding

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

where

$L_A(\mathrm{\Omega })=\{u\in M_f(\mathrm{\Omega }):\Vert u{\Vert }_{A,\mathrm{\Omega }}=\inf \{k>0:\underset{\mathrm{\Omega }}{\int }A\left(\frac{|u(x)|}{k}\right)dx\le 1\}<\mathrm{\infty }\}.$

The space

$L_A(\mathrm{\Omega })$

is an example of an Orlicz space.

References

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