Finite topological space: Difference between revisions
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In [[mathematics]], the '''Aubin–Lions lemma''' is a result in the theory of [[Sobolev space]]s of [[Banach space]]-valued functions. More precisely, it is a [[compact space|compactness]] criterion that is very useful in the study of nonlinear evolutionary [[partial differential equation]]s. The result is named after the [[France|French]] [[mathematician]]s [[Thierry Aubin]] and [[Jacques-Louis Lions]]. | |||
==Statement of the lemma== | |||
Let ''X''<sub>0</sub>, ''X'' and ''X''<sub>1</sub> be three Banach spaces with ''X''<sub>0</sub> ⊆ ''X'' ⊆ ''X''<sub>1</sub>. Suppose that ''X''<sub>0</sub> is [[compactly embedded]] in ''X'' and that ''X'' is [[continuously embedded]] in ''X''<sub>1</sub>; suppose also that ''X''<sub>0</sub> and ''X''<sub>1</sub> are [[reflexive space]]s. For 1 < ''p'', ''q'' < +∞, let | |||
:<math>W = \{ u \in L^p ([0, T]; X_0) | \dot{u} \in L^q ([0, T]; X_1) \}.</math> | |||
Then the embedding of ''W'' into ''L''<sup>''p''</sup>([0, ''T'']; ''X'') is also compact | |||
==References== | |||
* {{cite book | |||
| last = Showalter | |||
| first = Ralph E. | |||
| title = Monotone operators in Banach space and nonlinear partial differential equations | |||
| series = Mathematical Surveys and Monographs 49 | |||
| publisher = American Mathematical Society | |||
| location = Providence, RI | |||
| year = 1997 | |||
| pages = 106 | |||
| isbn = 0-8218-0500-2 | |||
| mr = 1422252 | |||
}} (Proposition III.1.3) | |||
{{DEFAULTSORT:Aubin-Lions lemma}} | |||
[[Category:Banach spaces]] | |||
[[Category:Functional analysis]] | |||
[[Category:Lemmas]] | |||
[[Category:Measure theory]] | |||
Revision as of 22:18, 18 January 2014
In mathematics, the Aubin–Lions lemma is a result in the theory of Sobolev spaces of Banach space-valued functions. More precisely, it is a compactness criterion that is very useful in the study of nonlinear evolutionary partial differential equations. The result is named after the French mathematicians Thierry Aubin and Jacques-Louis Lions.
Statement of the lemma
Let X0, X and X1 be three Banach spaces with X0 ⊆ X ⊆ X1. Suppose that X0 is compactly embedded in X and that X is continuously embedded in X1; suppose also that X0 and X1 are reflexive spaces. For 1 < p, q < +∞, let
![W=\{u\in L^{p}([0,T];X_{0})|{\dot {u}}\in L^{q}([0,T];X_{1})\}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebfc154a1a73bb58d0a664d30e23016911bc1abb)
Then the embedding of W into Lp([0, T]; X) is also compact
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 (Proposition III.1.3)