Sobolev inequality

For $1\leq p, define the Sobolev conjugate 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 $L^{p}(\Omega)$.

Theorem 1

Assume that $p\in[1,n)$ and let $\Omega$ be a bounded    , open subset of $\mathbb{R}^{n}$ with Lipschitz  boundary. Then there is a constant $C>0$ such that, for all $u\in W^{1,p}(\Omega)$ one has

 $\|u\|_{L^{p^{*}}(\Omega)}\leq C\|\nabla u\|_{L^{p}(\Omega)}.$

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   .

 Title Sobolev inequality Canonical name SobolevInequality Date of creation 2013-03-22 15:05:14 Last modified on 2013-03-22 15:05:14 Owner paolini (1187) Last modified by paolini (1187) Numerical id 10 Author paolini (1187) Entry type Theorem Classification msc 46E35 Synonym Sobolev embedding Synonym sobolev immersion Synonym Gagliardo Nirenberg inequality Related topic LpSpace Defines Sobolev conjugate Defines Sobolev exponent