A sharp weighted Wirtinger inequality

Tonia Ricciardi

Bollettino dell'Unione Matematica Italiana (2005)

  • Volume: 8-B, Issue: 1, page 259-267
  • ISSN: 0392-4041

Abstract

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We obtain a sharp estimate for the best constant C > 0 in the Wirtinger type inequality 0 2 π γ p ω 2 C 0 2 π γ q ω 2 where γ is bounded above and below away from zero, w is 2 π -periodic and such that 0 2 π γ p ω = 0 , and p + q 0 . Our result generalizes an inequality of Piccinini and Spagnolo.

How to cite

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Ricciardi, Tonia. "A sharp weighted Wirtinger inequality." Bollettino dell'Unione Matematica Italiana 8-B.1 (2005): 259-267. <http://eudml.org/doc/195264>.

@article{Ricciardi2005,
abstract = {We obtain a sharp estimate for the best constant $C>0$ in the Wirtinger type inequality $$ \int\_\{0\}^\{2\pi\}\gamma^\{p\}\omega^\{2\}\leq C \int\_\{0\}^\{2\pi\}\gamma^\{q\}\omega'^\{2\} $$ where $\gamma$ is bounded above and below away from zero, $w$ is $2\pi$-periodic and such that $\int_\{0\}^\{2\pi\}\gamma^\{p\}\omega=0$, and $p+q\geq 0$. Our result generalizes an inequality of Piccinini and Spagnolo.},
author = {Ricciardi, Tonia},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {259-267},
publisher = {Unione Matematica Italiana},
title = {A sharp weighted Wirtinger inequality},
url = {http://eudml.org/doc/195264},
volume = {8-B},
year = {2005},
}

TY - JOUR
AU - Ricciardi, Tonia
TI - A sharp weighted Wirtinger inequality
JO - Bollettino dell'Unione Matematica Italiana
DA - 2005/2//
PB - Unione Matematica Italiana
VL - 8-B
IS - 1
SP - 259
EP - 267
AB - We obtain a sharp estimate for the best constant $C>0$ in the Wirtinger type inequality $$ \int_{0}^{2\pi}\gamma^{p}\omega^{2}\leq C \int_{0}^{2\pi}\gamma^{q}\omega'^{2} $$ where $\gamma$ is bounded above and below away from zero, $w$ is $2\pi$-periodic and such that $\int_{0}^{2\pi}\gamma^{p}\omega=0$, and $p+q\geq 0$. Our result generalizes an inequality of Piccinini and Spagnolo.
LA - eng
UR - http://eudml.org/doc/195264
ER -

References

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  1. BEESACK, P. R., Integral inequalities of the Wirtinger type, Duke Math. Jour., 25 (1958), 477-498. Zbl0082.27104MR97478
  2. CROCE, G. - DACOROGNA, B., On a generalized Wirtinger inequality, Discrete Cont. Dynam. Systems, 9, No. 5 (2003), 1329-1341. Zbl1055.49033MR1974431
  3. DACOROGNA, B. - GANGBO, W. - SUBÍA, N., Sur une généralisation de l’inégalité de Wirtinger, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 29-50. Zbl0764.49009MR1151466
  4. PICCININI, L. C. - SPAGNOLO, S., On the Hölder continuity of solutions of second order elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa, 26, No. 2 (1972), 391-402. Zbl0237.35028MR361422

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