Inequality in a Sobolev space In the Sobolev space $H^2(\mathbb{R}^2)$ is it true that there exists a constant $c&gt;0$ 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.
So this is called the Poincaré inequality. It is true on a bounded domain. When you are in $\mathbb{R}^d$ and not in a bounded domain, you need some additional weights (for example it works with Gaussian weights, or sufficiently decaying weights).
In the form you are asking, it is false. indeed, take $\langle x\rangle = \sqrt{1+|x|^2}$ and
$$
u(x) = \frac{1}{\langle x\rangle}
$$
Then $u∉ L^2$ (since $u^2 ∼ |x|^{-2}$ when $x\to\infty$), but
$$
∇u  = \frac{-x}{\langle x\rangle^3}
$$
therefore $∇u ∈ L^2$ since $|∇u|^2 = \frac{|x|^2}{\langle x\rangle^6} ∼ |x|^{-4}$ when $x\to\infty$.
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