# Poincare's Inequality

Assume $$\Omega \subset L_d$$, for some $$d > 0$$. Then, for all $$u \in W^{1,q}_0(\Omega)$$ $$1 \leq q \leq \infty$$ , $$\left \|u \right \|_p \leq (d/2)\left \| \nabla u \right \|_p$$. Prove that the Inequality fails, in general, if $$\Omega$$ is not contained in some layer $$L_d$$. Suppose, for instance, $$\Omega$$$$R^n$$

• What is $L_d$?? – Jeff Jun 15 at 20:10