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Difference between revisions of "Poincaré inequality"

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===Poincaré inequality in ball (case  $1\leqslant p < n$)===
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Let $f\in W^1_p(\mathbb R^n)$, $1\leqslant p < n$ and $p^* = \frac{np}{n-p}$ then the following inequality holds
 
Let $f\in W^1_p(\mathbb R^n)$, $1\leqslant p < n$ and $p^* = \frac{np}{n-p}$ then the following inequality holds
 
\begin{equation}\label{eq:1}
 
\begin{equation}\label{eq:1}
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\end{equation}
 
\end{equation}
 
for any balls $B \subset \mathbb R^n$, and constant $C$ depends only on $n$ and $p$. Here $f_B = \frac{1}{|B|}\int\limits_{B}f\,dx$.  
 
for any balls $B \subset \mathbb R^n$, and constant $C$ depends only on $n$ and $p$. Here $f_B = \frac{1}{|B|}\int\limits_{B}f\,dx$.  
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===Poincaré inequality in ball (case  $1\leqslant p < \infty$)===
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There is a weaker inequality which is derived from \ref{eq:1} by inserting the measure of ball B and applying [[Hölder_inequality | Hölder inequality]].
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\begin{equation}\label{eq:2}
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\frac{1}{|B|}\int\limits_{B}|f(x)-f_B|^{p}\,dx \leqslant \frac{Cr^p}{|B|}\int\limits_{B}|\nabla f(x)|^{p}\,dx,
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\end{equation}
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where $r$ denotes the radius $B$.
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===References===
 
===References===
 
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Revision as of 19:27, 27 November 2012


Poincaré inequality in ball (case $1\leqslant p < n$)

Let $f\in W^1_p(\mathbb R^n)$, $1\leqslant p < n$ and $p^* = \frac{np}{n-p}$ then the following inequality holds \begin{equation}\label{eq:1} \Bigl(\int\limits_{B}|f(x)-f_B|^{p^*}\,dx\Bigr)^{\frac{1}{p^*}} \leqslant C\Bigl(\int\limits_{B}|\nabla f(x)|^{p}\,dx\Bigr)^{\frac{1}{p}} \end{equation} for any balls $B \subset \mathbb R^n$, and constant $C$ depends only on $n$ and $p$. Here $f_B = \frac{1}{|B|}\int\limits_{B}f\,dx$.

Poincaré inequality in ball (case $1\leqslant p < \infty$)

There is a weaker inequality which is derived from \ref{eq:1} by inserting the measure of ball B and applying Hölder inequality.

\begin{equation}\label{eq:2} \frac{1}{|B|}\int\limits_{B}|f(x)-f_B|^{p}\,dx \leqslant \frac{Cr^p}{|B|}\int\limits_{B}|\nabla f(x)|^{p}\,dx, \end{equation} where $r$ denotes the radius $B$.

References

[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[JH] J. Heinonen, "Lectures on Analysis on Metric Spaces" Springer, New York, NY, 2001.
How to Cite This Entry:
Poincaré inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_inequality&oldid=28918