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For $\nu\in (\epsilon,1)$ with $0<\epsilon<1$, let $P_\nu:H_0^1(\Omega)\rightarrow H_0^1(\Omega)$ with $\Omega\subset \mathbb{R}^N$ bounded Lipschitz domain, be the projection operator onto the set $K_\nu=\{v\in H_0^1(\Omega): |\nabla v|_{\mathbb{R}^N}\leq \nu \quad a.e.\}$. Is there an $L>0$ such that $|P_\nu w-P_\rho w|_{H_0^1(\Omega)}\leq L |\nu-\rho|,$ for all $w\in H_0^1(\Omega)$? Does the answer depend on $N$?

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    $Mercy, yes, I edited 40 minutes before the comment; it is strange that you saw the old version, though! Thanks anyway!2012-09-03

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