Inequality in a Sobolev space In the Sobolev space $H^2(\mathbb{R}^2)$ is it true that there exists a constant $c>0$ such that
$$\|u\|_{L^2(\mathbb{R}^2)}\le c\|\nabla u\|_{L^2(\mathbb{R}^2)}\;,\quad\quad \text{for all}\quad u\in H^2(\mathbb{R}^2)?$$
Thank you in advance.

No
$u=1|_{ [0,1]^2} $ indicatrice function of $ [0,1] ^2$
$u\in H^2(R^2)$
$\nabla u=0$, but $u $ no zero
