Is the constant of Poincaré inequality related with the measure of the set?

Question

If I'm not mistaken the constant of the Poincaré inequality is related to the measure of set. For example in a ball. I'd like that someone told me up or indicate a reference for me. I'll be grateful, thanks.

Answer 1

Poincaré inequality works if the open $\Omega$ set we are working with has a finite measure in a direction, namely if there is $v$ of norm $1$ such that $S:=\{\lambda,\lambda v\in\Omega\}$ has a finite measure. In this case, we have Poincaré inequality. Indeed, rotating $\Omega$, we can assume that $v=e_n$, then we show it for test functions: $$u(x)=\int_{-\infty}^{x_n}\partial_{x_n}u(x_1,\dots,x_{n-1},t)dt$$ hence $$\int_{\Omega}|u(x)|^2dx\leq |S|^2\int_{\Omega}|\nabla u|^2dx.$$ Then we conclude by density.