A sharp weighted Wirtinger inequality and some related functional spaces
Giova, Raffaella ; Ricciardi, Tonia
Bull. Belg. Math. Soc. Simon Stevin, Tome 17 (2010) no. 1, p. 209-218 / Harvested from Project Euclid
Text on Project Euclid
PDF
Alt PDF
We consider the generalized Wirtinger inequality \[ \left( \int_{0}^{T} a |u|^q \right)^{1/q} \le C \biggm(\int_{0}^{T} a^{1-p} |u'|^{p}\biggm)^{1/p}, \] with $p,q>1$, $T>0$, $a\in L^1[0,T]$, $a\ge0$, $a\not\equiv0$ and where $u$ is a $T$-periodic function satisfying the constraint \[ \int_{0}^{T} a |u|^{q-2}u =0. \] We provide the best constant $C>0$ as well as all extremals. Furthermore, we characterize the natural functional space where the inequality is defined.
Publié le : 2010-04-15
Classification:  Weighted Wirtinger inequality,  best constant,  weighted Sobolev space,  generalized trigonometric functions,  26D15 
@article{1274896200,
     author = {Giova, Raffaella and Ricciardi, Tonia},
     title = {A sharp weighted Wirtinger inequality and some related functional spaces},
     journal = {Bull. Belg. Math. Soc. Simon Stevin},
     volume = {17},
     number = {1},
     year = {2010},
     pages = { 209-218},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1274896200}
}

Giova, Raffaella; Ricciardi, Tonia. A sharp weighted Wirtinger inequality and some related functional spaces. Bull. Belg. Math. Soc. Simon Stevin, Tome 17 (2010) no. 1, pp.  209-218. http://gdmltest.u-ga.fr/item/1274896200/