Poincaré inequality for Lipschitz functions with bounded domain

Question

Let $u\in W^{1,\infty}(B_h(0),\mathbb R^n)$, where $B_h(0)=\{x\in\mathbb R^n:|x|<h\}$.
From the Poincaré inequality we know that $$ \|u-\mathrm{Id}-\frac{1}{\mathrm{Vol}(B_h(0))}\int_{B_h(0)}(u-\mathrm{Id})\|_{L^2} \leq C\|du-\mathrm{Id}\|_{L^2} $$ for some constant $C>0$ independent of $u$. Now I want to bound $\|u-\mathrm{Id}\|_{W^{1,2}}$ in terms of $du$.
Is it also true that there exists a constant $C'>0$ such that $$ \|u-\mathrm{Id}\|_{L^2}\leq C'\|du-\mathrm{Id}\|_{L^2} $$ so that $\|u-\mathrm{Id}\|_{W^{1,2}}\leq C''\|du-\mathrm{Id}\|_{L^2}$ independent of $u$?

Answer 1

Is is not true. Consider a constant function u(x)=N. The estimate would lead to a contradiction for a sufficiently large N.