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

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

Then there exists the embedding

${\displaystyle W^{m,p}(\Omega )\hookrightarrow L_{A}(\Omega )}$

where

${\displaystyle L_{A}(\Omega )=\left\{u\in M_{f}(\Omega ):\|u\|_{A,\Omega }=\inf\{k>0:\int _{\Omega }A\left({\frac {|u(x)|}{k}}\right)~dx\leq 1\}<\infty \right\}.}$

The space

${\displaystyle L_{A}(\Omega )}$

is an example of an Orlicz space.

References

• Moser, J. (1971), "A Sharp form of an Inequality by N. Trudinger", Indiana Univ. Math. J., 20 (11): 1077–1092, doi:10.1512/iumj.1971.20.20101.
• Trudinger, N. S. (1967), "On imbeddings into Orlicz spaces and some applications", J. Math. Mech., 17: 473–483.