September  2003, 9(5): 1329-1341. doi: 10.3934/dcds.2003.9.1329

On a generalized Wirtinger inequality

1. 

Département de Mathématiques, EPFL, 1015 Lausanne CH, Chile, Chile

Received  August 2002 Revised  January 2003 Published  June 2003

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.

Citation: Gisella Croce, Bernard Dacorogna. On a generalized Wirtinger inequality. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1329-1341. doi: 10.3934/dcds.2003.9.1329
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