# Poincaré inequality

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### 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.

\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 $B$.