Suppose $U\subseteq \mathbb R^n$ is bounded an has $C^1$ boundary. The extension theorem states that there exists a bounded linear operator $$E:W^{1,p}(U)\to W^{1,p}(\mathbb R^n)$$ such that $Eu$ has compact support and $Eu|_U=u$. Furthermore, $||Eu||_{W^{1,p}(\mathbb R^n)}\le C||u||_{W^{1,p}(U)}$.
My question is, do we have $||Eu||_{L^p(\mathbb R^n)}\le C||u||_{L^p(U)}$ and $||D(Eu)||_{L^p(\mathbb R^n)}\le C||Du||_{L^p(U)}$?
I doubt so, but I do not see where it fails from the proof that Evans gives in his PDE book.