[Math] general conditions for reverse poincare inequality

Question

I'd like to know when the reverse Poincare inequality is true: Given a bounded domain $\Omega$, and $f \in L^2(\Omega)$, under what conditions on $f$ (neglecting the trivial constant case) and/or $\Omega$ is it true that there is a constant $C(\Omega)$ such that $\|\nabla f\|_{L^2} \leq C \|f\|_{L^2}$?

Thanks in advance.

Answer 1

I don't think such a thing can be true without some extreme restrictions. Just take the simple domain $[0,1]$ and $f_n(x):=\frac{1}{n} \sin(n^2x)$ then you can see that $f_n\to 0$ in $L^2$ but $\|Df_n\|$ will diverge so the inequality can't hold.