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Given a smooth bounded open subset $\Omega$ of $\mathbb{R}^n$, does there exist $A >0$ such that if $f\in BV(\Omega)$ with zero trace on $\partial \Omega$, and $\int_\Omega |Df| = 1$, then $\|f\|_{L^1(\Omega)} \leq A$? This is not a homework problem.

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    You mean $A$ should be independent of the choice of $f \in BV(\Omega)$ ?2012-04-11
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    I found a possible reference: Integral Inequalities of Poincaré and Wirtinger Type for BV Functions Norman G. Meyers and William P. Ziemer American Journal of Mathematics Vol. 99, No. 6 (Dec., 1977), pp. 1345-13602012-04-11
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    Yes, A is independent of f2012-04-11

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