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Sobolev inequality
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(Theorem)
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For $1\le p < n$ , define the Sobolev conjugate 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 $L^p(\Omega)$ .
Theorem 1 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 $$ \Vert u \Vert_{L^{p^*}(\Omega)} \le C \Vert \nabla u \Vert_{L^p(\Omega)}. $$
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.
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"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 |
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Cross-references: continuous, inclusion map, subspace, theorem, boundary, Lipschitz, open subset, bounded, functions, Sobolev space, weak derivative, represent
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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 12349 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) |
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Pending Errata and Addenda
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