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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.

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up vote 3 down vote accepted

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.

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