Poincaré inequality

from The Encyclopedia of Mathematics

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

$$\begin{equation} \Bigl(\int\limits_{B}|f(x)-f_B|^{p^*}\,dx\Bigr)^{\frac{1}{p^*}} \leqslant C\Bigl(\int\limits_{B}|\nabla f(x)|^{p}\,dx\Bigr)^{\frac{1}{p}} \end{equation}$$

for any balls $B \subset \mathbb R^n$, and constant $C$ depends only on $n$ and $p$. Here $f_B = \frac{1}{|B|}\int\limits_{B}f\,dx$.

Lipschitz Function

from Encyclopedia of Mathematics

Начали писать статью про Липшецеву функцию.

Let a function $f:[a,b]\to \mathbb R$ be such that for some constant M and for all $x,y\in [a,b]$
$$\begin{equation} |f(x)-f(y)| \leq M|x-y|. \end{equation}$$
Then the function $f$ is called Lipschitz on $[a,b]$ , and one writes $f\in \operatorname{Lip}_M[a,b]$.

Generalized derivative

from Encyclopedia of Mathematics.

An extension of the idea of a derivative to some classes of non-differentiable functions. The first definition is due to S.L. Sobolev, who arrived at a definition of a generalized derivative from the point of view of his concept of a generalized function.