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| | I'm Shawnee (25) from Meadville, United States. <br>I'm learning German literature at a local high school and I'm just about to graduate.<br>I have a part time job in a college.<br><br>my page; [http://intlfa.com/index.php?do=/profile-6730/info/ wordpress dropbox backup] |
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| In [[mathematics]], '''Souček spaces''' are generalizations of [[Sobolev spaces]], named after the [[Czech people|Czech]] [[mathematician]] [[Jiří Souček]]. One of their main advantages is that they offer a way to deal with the fact that the Sobolev space ''W''<sup>1,1</sup> is not a [[reflexive space]]; since ''W''<sup>1,1</sup> is not reflexive, it is not always true that a bounded sequence has a [[weak topology|weakly convergent]] [[subsequence]], which is a desideratum in many applications.
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| ==Definition==
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| Let Ω be a [[bounded set|bounded domain]] in ''n''-dimensional [[Euclidean space]] with smooth [[boundary (topology)|boundary]]. The '''Souček space''' ''W''<sup>1,''μ''</sup>(Ω; '''R'''<sup>''m''</sup>) is defined to be the space of all [[ordered pair]]s (''u'', ''v''), where
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| * ''u'' lies in the [[Lp space|Lebesgue space]] ''L''<sup>1</sup>(Ω; '''R'''<sup>''m''</sup>);
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| * ''v'' (thought of as the gradient of ''u'') is a [[regular measure|regular]] [[Borel measure]] on the [[closure (topology)|closure]] of Ω;
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| * there exists a sequence of functions ''u''<sub>''k''</sub> in the Sobolev space ''W''<sup>1,1</sup>(Ω; '''R'''<sup>''m''</sup>) such that
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| ::<math>\lim_{k \to \infty} u_{k} = u \mbox{ in } L^{1} (\Omega; \mathbf{R}^{m})</math>
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| :and
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| ::<math>\lim_{k \to \infty} \nabla u_{k} = v</math>
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| :weakly-∗ in the space of all [[vector-valued measure|'''R'''<sup>''m''×''n''</sup>-valued]] regular Borel measures on the closure of Ω.
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| ==Properties==
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| * The Souček space ''W''<sup>1,''μ''</sup>(Ω; '''R'''<sup>''m''</sup>) is a [[Banach space]] when equipped with the [[norm (mathematics)|norm]] given by
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| ::<math>\| (u, v) \| := \| u \|_{L^{1}} + \| v \|_{M},</math> | |
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| :i.e. the sum of the ''L''<sup>1</sup> and [[total variation]] norms of the two components.
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| ==References== | |
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| * {{cite journal
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| | last = Souček
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| | first = Jiří
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| | title = Spaces of functions on domain Ω, whose ''k''-th derivatives are measures defined on Ω̅
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| | journal = Časopis Pěst. Mat.
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| | volume = 97
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| | year = 1972
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| | pages = 10–46, 94
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| | issn = 0528-2195
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| }} {{MathSciNet|id=0313798}}
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| {{DEFAULTSORT:Soucek space}}
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| [[Category:Banach spaces]]
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| [[Category:Sobolev spaces]]
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I'm Shawnee (25) from Meadville, United States.
I'm learning German literature at a local high school and I'm just about to graduate.
I have a part time job in a college.
my page; wordpress dropbox backup