# American Institute of Mathematical Sciences

May  2012, 11(3): 1363-1386. doi: 10.3934/cpaa.2012.11.1363

## Improving sharp Sobolev type inequalities by optimal remainder gradient norms

 1 Dipartimento di Matematica "U.Dini", Università di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy 2 Dipartimento di Matematica, Seconda Università di Napoli, Viale Lincoln 5, 81100 Caserta, Italy

Received  December 2010 Revised  March 2011 Published  December 2011

We are concerned with Sobolev type inequalities in $W^{1,p}_0(\Omega )$, $\Omega \subset R^n$, with optimal target norms and sharp constants. Admissible remainder terms depending on the gradient are characterized. As a consequence, the strongest possible remainder norm of the gradient is exhibited. Both the case when $p< n$ and the borderline case when $p = n$ are considered. Related Hardy inequalities with remainders are also derived.
Citation: Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363
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