Sobolev inequality SobolevInequality 2005-02-23 06:49:48 2007-06-29 04:01:19 Theorem Sobolev conjugate Sobolev exponent % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newtheorem{theorem}{Theorem} \newcommand{\R}{\mathbb R} For $1\le p < n$, define the \emph{Sobolev conjugate \PMlinkescapetext{exponent}} of $p$ as \[ p^* := \frac {np}{n-p}. \] Note that $-n/p^* = 1-n/p$. In the following statement $\nabla$ represent the weak derivative and $W^{1,p}(\Omega)$ is the Sobolev space of functions $u\in L^p(\Omega)$ whose weak derivative $\nabla u$ is itself in $L^p(\Omega)$. \begin{theorem} Assume that $p\in [1,n)$ and let $\Omega$ be a bounded, open subset of $\R^n$ with Lipschitz boundary. Then there is a constant $C>0$ such that, for all $u\in W^{1,p}(\Omega)$ one has \[ \Vert u \Vert_{L^{p^*}(\Omega)} \le C \Vert \nabla u \Vert_{L^p(\Omega)}. \] \end{theorem} We can restate the previous Theorem by saying that the Sobolev space $W^{1,p}(\Omega)$ is a subspace of the Lebesgue space $L^{p^*}(\Omega)$ and that the inclusion map $i\colon W^{1,p}(\Omega)\to L^{q^*}(\Omega)$ is continuous.