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[[File:Nehari.png|thumb|250px|Nehari variety]]


In the [[calculus of variations]], a branch of [[mathematics]], a '''Nehari manifold''' is a manifold of functions, whose definition is motivated by the work of  {{harvs|txt|authorlink=Zeev Nehari|first=Zeev|last=Nehari|year1=1960|year2=1961}}. It is a [[differential manifold]] associated to the [[Dirichlet problem]] for the semilinear [[elliptic partial differential equation]]


:<math> -\triangle u  = |u|^{p-1}u,\text{ with  }u\mid_{\partial \Omega} = 0.</math>
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Here Δ is the [[Laplacian]] on a bounded domain Ω in '''R'''<sup>''n''</sup>.
 
There are infinitely many solutions to this problem. Solutions are precisely the critical points for the energy functional
 
:<math>J(v) = \frac12\int_{\Omega}{|\nabla v|^2\,d\mu}-\frac1{p+1}\int_{\Omega}{|v|^{p+1}\,d\mu}</math>
 
on the [[Sobolev space]] {{nowrap|1=''H''{{su|p=1|b=0}}(&Omega;)}}. The Nehari manifold is defined to be the set of {{nowrap|1=''v''&nbsp;&isin;&nbsp;''H''{{su|p=1|b=0}}(&Omega;)}} such that
:<math>\|\nabla v\|^2_{L^2(\Omega)} = \|v\|^{p+1}_{L^{p+1}(\Omega)} > 0.</math>
Solutions to the original variational problem that lie in the Nehari manifold are (constrained) minimizers of the energy, and so [[direct methods in the calculus of variations]] can be brought to bear.
 
More generally, given a suitable functional ''J'', the associated Nehari manifold is defined as the set of functions ''u'' in an appropriate function space for which
 
:<math>\langle J'(u), u\rangle = 0. \, </math>
 
Here ''J''&prime; is the functional derivative of ''J''.
 
== References ==
* A. Bahri and P. L. Lions (1988), Morse Index of Some Min-Max Critical Points. I. Applications to Multiplicity Results. Communications on Pure and Applied Mathematics. (XLI) 1027&ndash;1037.
*{{Citation | last1=Nehari | first1=Zeev | title=On a class of nonlinear second-order differential equations | url=http://www.jstor.org/stable/1993333 | id={{MathSciNet | id = 0111898}} | year=1960 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=95 | pages=101–123}}
*{{Citation | last1=Nehari | first1=Zeev | title=Characteristic values associated with a class of non-linear second-order differential equations | doi= 10.1007/BF02559588 | id={{MathSciNet | id = 0123775}} | year=1961 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=105 | pages=141–175}}
*{{Citation | last1=Willem | first1=Michel | title=Minimax theorems | publisher=Birkhäuser Boston | location=Boston, MA | series=Progress in Nonlinear Differential Equations and their Applications, 24 | isbn=978-0-8176-3913-6 | id={{MathSciNet | id = 1400007}} | year=1996}}
 
[[Category:Calculus of variations]]
{{mathanalysis-stub}}

Latest revision as of 15:19, 14 November 2014


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