Abstract
Some problems concerning an exact constant in the generalized Poincare inequality \(\lambda _{pq} = \min \frac{{\left\| {y'} \right\|_{L_p [0,1]} }}{{\left\| y \right\|_{L_q [0,1]} }}\) with \(\bar y = \int\limits_0^1 {y(t)dt = 0} \) are discussed. In particular, the conjecture that the extremal of this problem is symmetric is confirmed. Bibliography: 7 titles.
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References
A. P. Buslaev, V. A. Kondrat´ev, and A. I. Nazarov, “On a family of extremal problems and related properties of an integral,” Mat. Zametki [in Russian] 64, No. 6, 830–838 (1998); English transl,: Math. Notes, 64, No. 5–6, 719–725 (1999).
B. Dacorogna, W. Gangbo, and N. Subia, “Sur une generalisation de l'inegalite de Wirtinger,” Ann. Inst. Henri Poincaré. Analyse Nonlin., 9, 29–50 (1992).
Y. V. Egorov, “On a Kondratiev problem,” C. R. Acad. Sci. Paris. Ser. I, 324, 503–507 (1997).
M. Belloni and B. Kawohl, “A symmetry problem related to Wirtinger's and Poincare's inequality,” J. Diff. Eq., 156, 211–218 (1999).
B. Kawohl, “Symmetry results for functions yelding best constants in Sobolev-type inequalities,” Discr. and Contin. Dynam. Systems, 6, No. 3, 683–690 (2000).
Ya, Yu. Nikitin, Asymptotic Efficiency of Nonparametric Tests [in Russian], Fizmatlit “Nauka”, Moscow (1995).
B. Kawohl, “Symmetry or not?” Math. Intell., 20, 16–22 (1998).
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Nazarov, A.I. On Exact Constant in the Generalized Poincare Inequality. Journal of Mathematical Sciences 112, 4029–4047 (2002). https://doi.org/10.1023/A:1020006108806
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DOI: https://doi.org/10.1023/A:1020006108806