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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 $u$?

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

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  • $\begingroup$ Yes you are right. Would it work if we assume $u(0)=0$? $\endgroup$ – Pink Panther Jun 28 at 20:38
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    $\begingroup$ 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. $\endgroup$ – StarBug Jun 28 at 20:45

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