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An optimal Poincaré inequality in for convex domains

Author(s): Gabriel Acosta; Ricardo G. Durán
Journal: Proc. Amer. Math. Soc. 132 (2004), 195-202.
MSC (2000): Primary 26D10
Posted: April 24, 2003
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Abstract: For convex domains $\Omega\subset\mathbb{R}^n$ with diameter we prove

$\begin{displaymath}\Vert u\Vert _{L^1(\omega)} \le \frac{d}{2} \Vert\nabla u\Vert _{L^1(\omega)} \end{displaymath}$

for any

with zero mean value on $\omega$ . We also show that the constant

in this inequality is optimal.

References:

1.: S. AGMON, Lectures on Elliptic Boundary Value Problems, Van Nostrand Company, 1965. MR 31:2504
2.: D. GILBARG, N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Springer Verlag, Berlin, 1983. MR 86c:35035
3.: G. H. HARDY, J. E. LITTLEWOOD, G. POLYA, Inequalities, Cambridge Univ. Press, Cambridge (1952). MR 13:727e
4.: L. E. PAYNE, H. F. WEINBERGER, An optimal Poincaré inequality for convex domains, Arch. Rat. Mech. Anal. 5, 286-292, 1960. MR 22:8198
5.: R. VERFÜRTH, A note on polynomial approximation in Sobolev spaces, Math. Model. Meth. Appl. Sci. 33, 715-719, 1999. MR 2000h:41016
6.: R. J. GARDNER, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. 39, 355-405, 2002. MR 2003f:26035

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Additional Information:

Gabriel Acosta
Affiliation: Instituto de Ciencias, Universidad Nacional de General Sarmiento, J. M. Gutierrez 1150, Los Polvorines, B1613GSX Provincia de Buenos Aires, Argentina
Email: gacosta@ungs.edu.ar

Ricardo G. Durán
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Email: rduran@dm.uba.ar

DOI: 10.1090/S0002-9939-03-07004-7
PII: S 0002-9939(03)07004-7
Keywords: Poincar\'e inequalities, weighted estimates
Received by editor(s): May 10, 2002
Received by editor(s) in revised form: September 10, 2002
Posted: April 24, 2003
Additional Notes: This work was supported by Universidad de Buenos Aires under grant TX048, ANPCyT under grant PICT 03-05009 and by CONICET under grant PIP 0660/98. The second author is a member of CONICET, Argentina
Communicated by: Andreas Seeger
Copyright of article: Copyright 2003, American Mathematical Society


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