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 ${\mathbb{ℝ}}^n$ 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 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

• 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.