Improving  sharp Sobolev type inequalities  by  optimal remaindergradient norms

Andrea Cianchi; Adele Ferone

doi:10.3934/cpaa.2012.11.1363

Article Contents

2012, Volume 11, Issue 3: 1363-1386. Doi: 10.3934/cpaa.2012.11.1363

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Improving sharp Sobolev type inequalities by optimal remainder gradient norms

Andrea Cianchi^1, and
Adele Ferone^2,

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

Received: November 30, 2010

Revised: February 28, 2011

Published: April 2012

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Abstract

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.

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Mathematics Subject Classification: Primary: 46E35, 46E30.

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