IUMJ

Title: Moser-Trudinger inequalities without boundary conditions and isoperimetric problems

Authors: Andrea Cianchi

Issue: Volume 54 (2005), Issue 3, 669-706

Abstract:

The best constant is exhibited in Trudinger's exponential inequality for functions from the Sobolev space $W^{1,n}(\Omega)$, with $\Omega\subset\mathbb{R}^{n}$ and $n\geq2$. This complements a classical result by Moser dealing with the subspace $W^{1,n}_0(\Omega)$. An extension to the borderline Lorentz-Sobolev spaces $W^1L^{n,q}(\Omega)$ with $1<q\leq\infty$ is also established. A key step in our proofs is an asymptotically sharp relative isoperimetric inequality for domains in $\mathbb{R}^{n}$.