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Poincaré inequality is given by $$\int_\Omega u^2\le C\int_\Omega|\nabla u|^2dx ,$$ where $\Omega$ is bounded open region in $\mathbb R^n$. However this inequality is not satisfied by all the function. Take for example a constant function $u=10$ in some region.

Happy to have have some discussions about it. Thanks for your help.

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    You didn't state the inequality correctly; see here: http://en.wikipedia.org/wiki/Poincaré_inequality2012-07-20
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    @ChristianBlatter : Sir , i am referring to this link:[http://mathreview.uwaterloo.ca/archive/voli/2/nica.pdf2012-07-20
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    In the quoted source the author refers to a certain Sobolev space which is the completion of $C_0^1(\Omega)$. Here the ${}_0$ means that compact support is assumed; in particular these functions vanish on the boundary.2012-07-20

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This inequality is, as you have shown by a simple example, not valid as stated. Things can be fixed up in various ways:

In the Wikipedia article on the Poincaré inequality it is assumed that the mean value $u_\Omega$ of $u$ on $\Omega$ is zero, resp., the left side of the inequality is replaced by $\int_\Omega|u-u_\Omega|^2\ {\rm d}x$.

In the quoted source the author refers to a certain Sobolev space which is the completion of $C_0^1(\Omega)$. Here the ${}_0$ means that compact support is assumed; in particular these functions vanish on the boundary.

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    @Christain Blatter : Is there any difference to assume that the mean to be zero ?? Is there a special condtion to be applied to assume that the mean is zero ?2012-07-22
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    @Theorem: It's an assumption on $u$, namely that $\int_\Omega u\ {\rm d}x=0$. This assumption eliminates your example $u(x)\equiv10$.2012-07-22