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 [math]\displaystyle{ \Omega }[/math] be a bounded domain in [math]\displaystyle{ \mathbb{R}^n }[/math] satisfying the cone condition. Let [math]\displaystyle{ mp=n }[/math] and [math]\displaystyle{ p\gt 1 }[/math]. Set

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

Then there exists the embedding

[math]\displaystyle{ W^{m,p}(\Omega)\hookrightarrow L_A(\Omega) }[/math]

where

[math]\displaystyle{ L_A(\Omega)=\left\{ u\in M_f(\Omega):\|u\|_{A,\Omega}=\inf\{ k\gt 0:\int_\Omega A\left( \frac{|u(x)|}{k} \right)~dx\leq 1 \}\lt \infty \right\}. }[/math]

The space

[math]\displaystyle{ L_A(\Omega) }[/math]

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 .