PlanetMath: Sobolev inequality

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Sobolev inequality

(Theorem)

For $1\le p < n$ , define the Sobolev conjugate exponent of as

$\displaystyle p^* := \frac {np}{n-p}.$

Note that

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 such that, for all $u\in W^{1,p}(\Omega)$ one has

$\displaystyle \Vert u \Vert_{L^{p^*}(\Omega)} \le C \Vert \nabla u \Vert_{L^p(\Omega)}.$

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.

"Sobolev inequality" is owned by paolini.

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See Also:

-space

Other names:

Sobolev embedding, sobolev immersion, Gagliardo Nirenberg inequality

Also defines:

Sobolev conjugate, Sobolev exponent

Attachments:: proof of Sobolev inequality for $\Omega=\mathbf{R}^n$ (Proof) by vanschaf

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Cross-references: continuous, inclusion map, subspace, boundary, Lipschitz, open subset, bounded, functions, Sobolev space, weak derivative, represent
There is 1 reference to this entry.

This is version 7 of Sobolev inequality, born on 2005-02-23, modified 2007-06-29.
Object id is 6812, canonical name is SobolevInequality.
Accessed 8612 times total.

Classification:

AMS MSC:

46E35 (Functional analysis :: Linear function spaces and their duals :: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems)

Pending Errata and Addenda

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