Dacorogna, B.; Gangbo, W.; Subía, N. Sur une généralisation de l’inégalité de Wirtinger. (A generalization of Wirtinger’s inequality). (French) Zbl 0764.49009 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 9, No. 1, 29-50 (1992). Let \(\alpha_ 1=\alpha_ 1(p,q)=\min\left\{{\| u'\|_{L^ p}\over\| u\|_{L^ q}}| u\in W^{1,p}(- 1,1)\backslash\{0\},\;u(-1)=u(1),\;\int^ 1_{-1}u| u|^{q- 2}=0\right\}\), \(\alpha_ 2=\alpha_ 2(p,q)=\min\left\{{\| u'\|_{L^ p}\over\| u\|_{L^ q}}| u\in W^{1,p}(- 1,1)\backslash\{0\},\;u(-1)=u(1),\;\int^ 1_{-1}u=0\right\}\). We compute explicitly \(\alpha_ 1\) and we show that for \(q\leq 2p\), \(\alpha_ 1=\alpha_ 2\), while for \(q\) sufficiently large \(\alpha_ 2<\alpha_ 1\). Reviewer: B.Dacorogna (Lausanne) Cited in 1 ReviewCited in 22 Documents MSC: 49J40 Variational inequalities Keywords:calculus of variations; crystals; best Sobolev constant; Wirtinger’s inequality PDFBibTeX XMLCite \textit{B. Dacorogna} et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 9, No. 1, 29--50 (1992; Zbl 0764.49009) Full Text: DOI Numdam EuDML References: [1] Talenti, G., Best constant in Sobolev inequality, Ann. Mat. Pura Appl., vol. 110, 353-372 (1976) · Zbl 0353.46018 [2] Talenti, G., Calcolo delle variazioni (1977), Pitagora Editrice: Pitagora Editrice Bologna · Zbl 1308.49001 [3] Dacorogna, B.; Pfister, C. E., Wulff theorem and best constant in Sobolev inequality, J. Math. Pures Appl., vol. 71 (1992) · Zbl 0676.46031 [4] Wulff, G., Zur Frage der Geschwindigkeit des Wachsturms und der Auflösung der Kristallflächen, Z. Kristallogr., vol. 34, 449-530 (1901) [5] Dinghas, A., Über einen geometrischen satz von Wulff für die Gleichgewichts form von Kristallen, Z. Kristallogr., vol. 105, 304-314 (1944) · Zbl 0028.43001 [6] Taylor, J. E., Crystalline variational problems, Bull. A.M.S., vol. 84, 568-588 (1978) · Zbl 0392.49022 [7] Hardy, G. H.; Littlewood, J. E.; Polya, G., Inequalities (1961), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0634.26008 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.