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 $N$?
Is this family of projections $\nu\mapsto P_\nu$ Lipschitz continuous?
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real-analysis
functional-analysis
sobolev-spaces
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0$Mercy, yes, I edited 40 minutes before the comment; it is strange that you saw the old version, though! Thanks anyway! – 2012-09-03