View source for Poincaré inequality

===Poincaré inequality in a 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 ball $B \subset \mathbb R^n$, and the constant $C$ depends only on $n$ and $p$. Here $f_B = \frac{1}{|B|}\int\limits_{B}f\,dx$.

===Poincaré inequality in a 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 | 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 of $B$.

===References===
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|valign="top"|{{Ref|EG}}||  L.C. Evans, R.F. Gariepy,  "Measure theory   and fine properties of  functions" Studies in Advanced  Mathematics.  CRC  Press, Boca Raton,  FL,  1992. {{MR|1158660}}  {{ZBL|0804.2800}} 
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|valign="top"|{{Ref|JH}}||  J. Heinonen,  "Lectures on Analysis on Metric Spaces" Springer, New York, NY, 2001.
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