Inequality in a Sobolev space

Question

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.

Answer 1

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

Answer 2

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 $x\to\infty$.