I can't seem to find this anywhere but I can find that there are (at least) two definitions of the Poincare inequality. One is
$$
\int_\Omega |f|^2 dx \leq c\int_\Omega |\nabla f(x)|^2 dx
$$
for $f \in H_0^1(\Omega)$, and the other is
$$
\int_\Omega |f – \bar{f}|^2 dx \leq c\int_\Omega |\nabla f(x)|^2 dx
$$
where $\bar{f} = \int_\Omega f dx$ and $f \in H^1(\Omega)$. Here $\Omega \subset \mathbb R^n$ is say a nice bounded domain.
Should I be thinking of these as different (but called the same)? Or are they equivalent in some way that I can't see? For instance I would imagine that to prove the first implies the second I would say something like if $f \in H^1 \setminus H_0^1$ then if I subtract the mean $\bar{f}$ then $f – \bar{f} \in H_0^1$ but this just doesn't seem true. But then the fact they are called the same thing would seem to suggest that the two are equivalent or perhaps that one implies the other? Even if they are not, is there any relationship between the two?
Best Answer
The second inequality can be written as $$ \|f\|_{L^2(\Omega)}^2 \le c \|\nabla f\|_{L^2(\Omega)}^2 \quad \forall f\in H^1(\Omega): \ \int_\Omega f =0. $$
They are two different kind of similar inequalities: these allow to control the $L^2$-norm of the function by the $L^2$-norm of its gradient. Of course, constant functions do not satisfy such an inequality. So this case has to be ruled out either by the choice of a subspace of $H^1(\Omega)$ or by using $f-\bar f$ in the estimate on the left-hand side.