Regarding the Poincare Friedrichs inequality

Question

I am working on a two part problem. Part 1 was to prove the Poincare-Friedrichs inequality for n=1:

$\int_{0}^{\alpha} |f(t)|^2 dt \le C\int_{0}^{\alpha} |f'(t)|^2 dt$

for some constant $C$. I managed to do this using the Cauchy-Schwarz inequality. Part 2 is to show there does not exist a constant $C > 0$ such that

$\int_{0}^{\alpha} |f'(t)|^2 dt \le C\int_{0}^{\alpha} |f(t)|^2 dt$.

It's been a while since I've done proofs of this sort and I am not sure what is the best strategy for proving non-existence. Can anyone point me in the right direction?

Sorry, $\alpha$ is a positive scalar and $f$ is a continuously differentiable function on $[0,\alpha]$ such that $f(0)=0$. — fideo, Oct 10 '14 at 06:44
look here:http://math.stackexchange.com/questions/545656/prove-friedrichs-inequality — , Oct 10 '14 at 09:46

score 1 · Accepted Answer · answered Oct 10 '14 at 09:54

1

For the second part take $f_m (\xi )= \xi ^m$ then we have $$\frac{n}{2}=\int_0^1 |f_n'(x)|^2 dx\leq C\int_0^1 |f_n(x)|^2 dx =\frac{C}{2n+1} \mbox{for all } n\in\mathbb{N}$$ which is impossible.

answered Oct 10 '14 at 09:54

Thank you. So it looks like to prove nonexistence the best technique is to assume existence and reach a contradiction? – fideo Oct 10 '14 at 14:28

Regarding the Poincare Friedrichs inequality

1 Answers1