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An optimal Poincaré inequality in $L^1$ for convex domains

Authors: Gabriel Acosta and Ricardo G. Durán
Journal: Proc. Amer. Math. Soc. 132 (2004), 195-202
MSC (2000): Primary 26D10
Published electronically: April 24, 2003
MathSciNet review: 2021262
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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.

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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

Ricardo G. Durán
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
ORCID: 0000-0003-1349-3708

Keywords: Poincaré inequalities, weighted estimates
Received by editor(s): May 10, 2002
Received by editor(s) in revised form: September 10, 2002
Published electronically: 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
Article copyright: © Copyright 2003 American Mathematical Society