Fama–French three-factor model: Difference between revisions

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{{DISPLAYTITLE:''p''-Laplacian}}
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In [[mathematics]], the '''''p''-Laplacian''', or the '''''p''-Laplace operator''', is a quasilinear [[elliptic operator|elliptic]] [[partial differential operator]] of 2nd order. It is a generalization of the [[Laplace operator]], where <math>p</math> is allowed to range over <math>1 < p < \infty</math>. It is written as


:<math>\nabla \cdot (|\nabla u|^{p-2} \nabla u).</math>
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In the special case when <math>p=2</math>, it is the regular [[Laplacian]]. <ref name="E356">Evans, pp 356.</ref>
 
== Energy formulation ==
The solution of the ''p''-Laplace equation with [[Dirichlet boundary conditions]]
 
:<math>\nabla \cdot (|\nabla u|^{p-2} \nabla u) = 0</math>
 
in a domain <math>\Omega</math> is the minimizer of the [[energy functional]]
 
:<math>J(u) = \int |\nabla u|^p \,dx</math>
 
among all functions in the [[Sobolev space]] <math>W^{1,p}(\Omega)</math> satisfying the boundary conditions in the [[Trace operator|trace]] sense.<ref name="E356"/>
 
== Notes ==
{{reflist}}
== Sources ==
*{{cite journal | last = Evans | first = Lawrence C. | authorlink = Lawrence C. Evans | title = A New Proof of Local <math>C^{1,\alpha}</math> Regularity for Solutions of Certain Degenerate Elliptic P.D.E. | journal = [[Journal of Differential Equations]] | volume = 45 | pages = 356–373 | year = 1982| mr=672713}}
*{{cite journal | last = Lewis | first= John L. | title = Capacitary functions in convex rings | journal = [[Archive for Rational Mechanics and Analysis]] | volume = 66 | pages = 201–224 | year = 1977|mr=0477094}}
==Further reading==
*{{Citation
  | last = Ladyženskaja
  | first = O. A.
  | author-link = Olga Aleksandrovna Ladyzhenskaya
  | last2 = Solonnikov
  | first2 = V. A.
  | author2-link = Vsevold Solonnikov
  | last3 = Ural'ceva
  | first3 = N. N.
  | author3-link = Nina Uralt'seva
  | title = Linear and quasi-linear equations of parabolic type
  | place = Providence, RI
  | publisher = [[American Mathematical Society]]
  | series = Translations of Mathematical Monographs
  | volume = 23
  | year = 1968
  | pages = XI+648
  | language =
  | url = http://books.google.com/books?id=dolUcRSDPgkC&printsec=frontcover#v=onepage&q&f=true
  | doi =
  | id =
  | isbn =
  | mr = 0241821
  | zbl = 0174.15403
}}.
*{{cite journal | last = Uhlenbeck | first= K. |author-link= Karen Uhlenbeck | title = Regularity for a class of non-linear elliptic systems | journal = [[Acta Mathematica]] | volume = 138 | pages = 219-240 | year = 1977|mr=0474389}}
 
*[http://www.math.ntnu.no/~lqvist/p-laplace.pdf Notes on the p-Laplace equation] by Peter Lindqvist
{{mathanalysis-stub}}
 
[[Category:Elliptic partial differential equations]]

Revision as of 17:06, 19 February 2014

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