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I am reading POincare inequality from Evans ( can be found here : ) and i am having some trouble understanding it . I would be glad if someone could explain the renormalizing step ? Specially i didn't understand how average of $v_k=0$ and few steps there .

I am also interested to know under what conditions does Poincare inequality hold ? Suppose if i take a constant function then it doesn't hold .

Thank you

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Oh ok , i think i have got the answer to the the first question . if $\bar u$ is the mean of u and $v_k$ is the mean of $u-\bar u$ then the $v_k=0$ – Theorem Jun 15 '12 at 15:38

See the book "Applied Functional Analysis" by Zeidler on p. 136. Maybe that will help you.

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