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{{distinguish|Pigeonhole principle}} {{distinguish|Pigeonhole principle}}
In ], '''Dirichlet's principle''' in ] states that, if the function <math> u ( x ) </math> is the solution to ] In ], and particularly in ], '''Dirichlet's principle'''is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.

==Formal statement==
'''Dirichlet's principle''' states that, if the function <math> u ( x ) </math> is the solution to ]


:<math>\Delta u + f = 0\,</math> :<math>\Delta u + f = 0\,</math>
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amongst all twice differentiable functions <math>v</math> such that <math>v=g</math> on <math>\partial\Omega</math> (provided that there exists at least one function making the Dirichlet's integral finite). This concept is named after the German mathematician ]. amongst all twice differentiable functions <math>v</math> such that <math>v=g</math> on <math>\partial\Omega</math> (provided that there exists at least one function making the Dirichlet's integral finite). This concept is named after the German mathematician ].


==History==
Since the Dirichlet's integral is bounded from below, the existence of an ] is guaranteed. That this infimum is attained was taken for granted by ] (who coined the term ''Dirichlet's principle'') and others until ] gave an example of a functional that does not attain its minimum. ] later justified Riemann's use of Dirichlet's principle. Since the Dirichlet's integral is bounded from below, the existence of an ] is guaranteed. That this infimum is attained was taken for granted by ] (who coined the term ''Dirichlet's principle'') and others until ] gave an example of a functional that does not attain its minimum. ] later justified Riemann's use of Dirichlet's principle.


Revision as of 14:01, 23 November 2014

Not to be confused with Pigeonhole principle.

In mathematics, and particularly in potential theory, Dirichlet's principleis the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.

Formal statement

Dirichlet's principle states that, if the function 
  
    
      
        u
        (
        x
        )
      
    
    {\displaystyle u(x)}
  
 is the solution to Poisson's equation
Δ u + f = 0 {\displaystyle \Delta u+f=0\,}

on a domain Ω {\displaystyle \Omega } of R n {\displaystyle \mathbb {R} ^{n}} with boundary condition

u = g  on  ∂ Ω , {\displaystyle u=g{\text{ on }}\partial \Omega ,\,}

then u can be obtained as the minimizer of the Dirichlet's energy

E [ v ( x ) ] = ∫ Ω ( 1 2 | ∇ v | 2 − v f ) d x {\displaystyle E=\int _{\Omega }\left({\frac {1}{2}}|\nabla v|^{2}-vf\right)\,\mathrm {d} x}

amongst all twice differentiable functions v {\displaystyle v} such that v = g {\displaystyle v=g} on ∂ Ω {\displaystyle \partial \Omega } (provided that there exists at least one function making the Dirichlet's integral finite). This concept is named after the German mathematician Peter Gustav Lejeune Dirichlet.

History

Since the Dirichlet's integral is bounded from below, the existence of an infimum is guaranteed. That this infimum is attained was taken for granted by Riemann (who coined the term Dirichlet's principle) and others until Weierstrass gave an example of a functional that does not attain its minimum. Hilbert later justified Riemann's use of Dirichlet's principle.

See also

References

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