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In [[mathematics]], '''Nemytskii operators''' are a class of [[nonlinear operator]]s on [[Lp space|''L''<sup>''p''</sup> spaces]] with good [[continuous function|continuity]] and [[bounded function|boundedness]] properties. They take their name from the [[mathematician]] [[Viktor Vladimirovich Nemytskii]].
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Let &Omega; be a domain (an [[open set|open]] and [[connected space|connected set]]) in ''n''-dimensional [[Euclidean space]]. A function ''f''&nbsp;:&nbsp;&Omega;&nbsp;&times;&nbsp;'''R'''<sup>''m''</sup>&nbsp;&rarr;&nbsp;'''R''' is said to satisfy the '''Carathéodory conditions''' if
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* ''f''(''x'',&nbsp;''u'') is a continuous function of ''u'' for [[almost all]] ''x''&nbsp;&isin;&nbsp;&Omega;;
* ''f''(''x'',&nbsp;''u'') is a [[measurable function]] of ''x'' for all ''u''&nbsp;&isin;&nbsp;'''R'''<sup>''m''</sup>.


Given a function ''f'' satisfying the Carathéodory conditions and a function ''u''&nbsp;:&nbsp;&Omega;&nbsp;&rarr;&nbsp;'''R'''<sup>''m''</sup>, define a new function ''F''(''u'')&nbsp;:&nbsp;&Omega;&nbsp;&rarr;&nbsp;'''R''' by
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:<math>F(u)(x) = f \big( x, u(x) \big).</math>
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The function ''F'' is called a '''Nemytskii operator'''.
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==Boundedness theorem==
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Let &Omega; be a domain, let 1&nbsp;&lt;&nbsp;''p''&nbsp;&lt;&nbsp;+&infin; and let ''g''&nbsp;&isin;&nbsp;''L''<sup>''q''</sup>(&Omega;;&nbsp;'''R'''), with
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:<math>\frac1{p} + \frac1{q} = 1.</math>
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Suppose that ''f'' satisfies the Carathéodory conditions and that, for some constant ''C'' and all ''x'' and ''u'',
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:<math>\big| f(x, u) \big| \leq C | u |^{p - 1} + g(x).</math>
  <li>[http://enseignement-lsf.com/spip.php?article64#forum25246010 http://enseignement-lsf.com/spip.php?article64#forum25246010]</li>
 
 
Then the Nemytskii operator ''F'' as defined above is a bounded and continuous map from ''L''<sup>''p''</sup>(&Omega;;&nbsp;'''R'''<sup>''m''</sup>) into ''L''<sup>''q''</sup>(&Omega;;&nbsp;'''R''').
</ul>
 
==References==
 
* {{cite book
|   author = Renardy, Michael and Rogers, Robert C.
|   title = An introduction to partial differential equations
|   series = Texts in Applied Mathematics 13
|  edition = Second edition
|publisher = Springer-Verlag
| location = New York
|    year = 2004
|   pages = 356
|      id = ISBN 0-387-00444-0
}} (Section 9.3.4)
 
[[Category:Operator theory]]

Latest revision as of 21:26, 3 January 2015

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