An optimal Poincaré inequality in 𝐿¹ for convex domains

Acosta, Gabriel; Durán, Ricardo

doi:10.1090/S0002-9939-03-07004-7

An optimal Poincaré inequality in $L^1$ for convex domains
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by Gabriel Acosta and Ricardo G. Durán

Proc. Amer. Math. Soc. 132 (2004), 195-202

DOI: https://doi.org/10.1090/S0002-9939-03-07004-7

Published electronically: April 24, 2003

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Abstract:

For convex domains $\Omega \subset \mathbb {R}^n$ with diameter $d$ we prove \[ \|u\|_{L^1(\omega )} \le \frac {d}{2} \|\nabla u\|_{L^1(\omega )} \] for any $u$ with zero mean value on $\omega$. We also show that the constant $1/2$ in this inequality is optimal.

References

Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), 286–292 (1960). MR 117419, DOI 10.1007/BF00252910
Rüdiger Verfürth, A note on polynomial approximation in Sobolev spaces, M2AN Math. Model. Numer. Anal. 33 (1999), no. 4, 715–719 (English, with English and French summaries). MR 1726481, DOI 10.1051/m2an:1999159
R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 355–405. MR 1898210, DOI 10.1090/S0273-0979-02-00941-2