# Poincaré inequality for Lipschitz functions with bounded domain

Let $$u\in W^{1,\infty}(B_h(0),\mathbb R^n)$$, where $$B_h(0)=\{x\in\mathbb R^n:|x|.
From the Poincaré inequality we know that $$\|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$$?

## 1 Answer

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

• Yes you are right. Would it work if we assume $u(0)=0$? – Pink Panther Jun 28 '19 at 20:38
• Yes. But you have to be careful prescribing pointwise values of functions in Sobolev spaces. In the case $W^{1,\infty}$ you should be OK though, since this space imbeds into the space of continuous functions. – StarBug Jun 28 '19 at 20:45