PDE satisfied by projection of a function on Subspace

Problem's Description

Given an open bounded set $D\subset \mathbb R^N$, let $f\in W^{-1,q}(D)$ and let $u$ be a Sobolev function $u\in W_0^{1,p}(D)$ such that $u$ solve the PDE $$ \begin{cases} -\Delta_p u=f\;\text{in $D$}\\ u\in W_0^{1,p}(D) \end{cases} $$

Given $\Omega\subset D$ (can be assumed open, or quasi open) If we define $P_{\Omega}:W_0^{1,p}(D)\to W_0^{1,p}(\Omega)$, $P_{\Omega}=Proj_{W_0^{1,p}(\Omega)}$ (projection of the function to the subspace $W_0^{1,p}(\Omega)$)

I wish to prove that the function $u_{\Omega}=P_{\Omega}u$ solve the PDE

$$ \begin{cases} -\Delta_p u_{\Omega}=f\;\text{in $\Omega$}\\ u_{\Omega}\in W_0^{1,p}(\Omega) \end{cases} $$

It will also sufficient to me if I could find a proof just for $f\equiv 1$.

1. Book Variational principles and free-boundary problems

   • Avner Friedman