# Is this family of projections $\nu\mapsto P_\nu$ Lipschitz continuous?

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 $N$?

-
As it is now the set $K_\nu$ is not well defined. I guess what you meant is $$K_\nu=\{u \in H_0^1(\Omega):\ |\nabla u|_{\mathbb{R}^N}\le \nu \ \text{ a.e. } \}.$$ –  Mercy Sep 3 '12 at 0:26
\$Mercy, yes, I edited 40 minutes before the comment; it is strange that you saw the old version, though! Thanks anyway! –  Nonliapunov Sep 3 '12 at 0:30