Difference between revisions of "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 | 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} | ||
\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}} | \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} | \end{equation} | ||
| − | for any | + | 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=== | ===References=== | ||
{| | {| | ||
Latest revision as of 15:44, 1 September 2020
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 $B$.
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. |
Poincaré inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_inequality&oldid=28906