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 PDF

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

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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