Inequality for proof of Density Theorem Someone could help me white this question or indicate some reference?
Lemma:For any $\epsilon>0$  There exits a $C=C(n,\epsilon)$ such that for $u \in H^{1}(B_{1})$ with 
$|\{ x \in B_{1} ; u=0\}| \geq \epsilon|B_{1}|$
there holds
$\int\limits_{B_{1}} u^{2} \leq C\int\limits_{B_{1}} |Du|^{2}$
Here:
 $B_{1}$ is unit open ball
$|.|$ is a Lebesgue measure of $"."$
Thanks a lot 
 A: This follows immediately from Poincare's inequality:

Let $1 \le p < \infty$ and let $\Omega \subset \mathbb{R}^N$ be a connected extension domain for $W^{1,p}(\Omega)$ with finite measure. Let $E \subset \Omega$ be a Lebesgue measurable set with positive measure. Then there exists a constant $C = C(p,\Omega,E) > 0$ such that for all $u \in W^{1,p}(\Omega)$, $$\int_{\Omega}|u(x) - u_E|^p\,dx \le C\int_{\Omega}|\nabla u|^p\,dx,$$ where $u_E = \frac{1}{|E|}\int_Eu(x)\,dx$.

Notice that you can apply this with $p = 2$, $\Omega = B_1$ and $E =\{x \in B_1 : u = 0\}$. For a proof of the result you can take a look at Leoni's book on Sobolev spaces. If you want to prove it by yourself argue by contradiction that such a constant does not exist and find a sequence for which the opposite inequality is satisfied for every $n$. After renormalizing the sequence find a convergent subsequence, apply Rellich-Kondrachov and find a contradiction. The fact the $E$ has positive measure comes into play at the end to rule out constants different from $0$.
