inequality for the projection of functions on convex sets with pointwise constraint

Hello everybody!

Take the following convex set which is a subset of the sobolev space
H^1_0(\Omega)

K=\{ v\in H^1_0(\Omega) s.t. \forall x\in\Omega, \vert Dv(x)\vert
\leq 1 \}

take any function from H^1_0(\Omega)

u\in H^1_0(\Omega)

and lets denote the projection of $u$ on $K$ by $P_K(u)$,
that is $P_K(u)$ is the nearest function in $K$ to the function $u$ in
the norm of $H^1_0(\Omega)$.

I am interested if such an inequality holds

\int_{\Omega}\vert u-P_K(u)\vert^2 dx \leq C \int_{\vert Du(x)
\vert > 1}\vert u-P_K(u)\vert^2 dx

here the constant $C$ may depend on the norm of $u$.

thanks in advance for any information.

kar

.

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