Moser-Trudinger inequalities without boundary conditions and isoperimetric problems Andrea Cianchi 46E3546E30Sobolev inequalitiessharp constantsrelative isoperimetric inequalitesrearrangements 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 $\mathbb{R}^{n}$. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2589 10.1512/iumj.2005.54.2589 en Indiana Univ. Math. J. 54 (2005) 669 - 706 state-of-the-art mathematics http://iumj.org/access/