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I am reading POincare inequality from Evans ( can be found here : http://anhngq.wordpress.com/2010/02/22/the-poincare-inequality/ ) 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$2012-06-15

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

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Pay attention that if a function $u$ is constant (ie. $\nabla u=0 $) and is also in the Sobolev space $W^{1,p}_0(\Omega)$, then the function must be equal to zero almost everywhere (actually everywhere if we take its continuous representative, if we are consistent with the notation of Brezi's Book) in order to satisfy the boundary conditions. (By a well known theorem in fact the closure of the space of differentailble functions in the sobolev space $W^{1,p}(\Omega)$, that we denote by $W^{1,p}_0(\Omega)$, is nothing but the functions in $W^{1,p}(\Omega)$ that are equal to zero on the boundary $\partial \Omega$)

Then

$\|u\|_{L^{p}(\Omega )}\leq C\|\nabla u\|_{L^{p}(\Omega )},$

becomes the equality $0=C\cdot 0$ where the constant $C$ is not relevant

That's it