Poincaré inequality using $H^1$ seminorm

Question

Does this inequality holds for Poincaré Inequality?

$$\|v\|_{L^2} \leqslant C_p |v|_{H^1}$$

and $$ |v|_{H^1} = \|v'\|_{L^2} $$ where $| \cdot |$ denotes the semi norm and $\|\cdot\|$ the norm.

I'm really confused with norms and semi norms in $H^1$ and $L^2$.

Cheers

Answer 1

The answer is no, which you can verify by calculating $\|v\|_{L^2}$ and $\|v'\|_{L^2}$ for the function $v_n(x)=\min(1,\max(0,n-|x|))$. As $n\to\infty$, one of the norms grows indefinitely while the other remains constant.