On a generalized Wirtinger inequality

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Received: July 31, 2002

Revised: December 31, 2002

Published: August 2003

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Abstract

Let

$\alpha( p,q,r) =$inf{$\frac{|| u'||_p}{||u||_q}:u\in W_{p e r}^{1,p}( -1,1) $\{$ 0$}, $\int_{-1}^1|u|^{r-2} u=0$} .

We show that

$\alpha( p,q,r )=\alpha ( p,q,q)$ if $q\leq rp+r-1$

$\alpha( p,q,r) <\alpha( p,q,q) $ if $q> ( 2r-1) p$

generalizing results of Dacorogna-Gangbo-Subía and others.

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Mathematics Subject Classification: 49R50, 26D10.

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