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In ], ''']'s principle''' in ] states that the ] <math>u</math> on a ] <math>\Omega</math> with ] In ], '''Dirichlet's principle''' in ] states that, if the function <math>u(x)</math> is the solution to Poisson's equation


:<math>u=g</math> on <math>\partial\Omega</math> :<math>\Delta u + f = 0</math>


on a ] <math>\Omega</math> of <math>\mathbb{R}^n</math> with ]
⚫ can be obtained as the minimizer of the ]


:<math>\int_\Omega |\nabla v|^2</math> :<math>u=g</math> on <math>\partial\Omega</math>,


⚫ then <math>u</math> can be obtained as the minimizer of the ]
amongst all functions


:<math>v</math> such that <math>v=g</math> on <math>\partial\Omega</math>, :<math>E = \int_\Omega \left(\frac{1}{2}|\nabla v|^2 - vf\right)\mathrm{d}x</math>


provided only that there exists one such function making the Dirichlet integral finite. amongst all twice differentiable functions <math>v</math> such that <math>v=g</math> on <math>\partial\Omega</math>, provided only that there exists one such function making the Dirichlet integral finite. This concept is named after the German mathematician ].


Since the Dirichlet integral is nonnegative, 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 Since the Dirichlet integral is nonnegative, 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
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==References== ==References==
* {{cite book | author=Lawrence C. Evans | title=Partial Differential Equations | publisher=American Mathematical Society | year=1998 | id=ISBN 978-0821807729 }}
* {{MathWorld | urlname=DirichletsPrinciple | title=Dirichlet's Principle}} * {{MathWorld | urlname=DirichletsPrinciple | title=Dirichlet's Principle}}


Revision as of 22:16, 30 July 2008

In mathematics, Dirichlet's principle in potential theory 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 {\displaystyle u=g} on ∂ Ω {\displaystyle \partial \Omega } ,

then u {\displaystyle u} can be obtained as the minimizer of the Dirichlet 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 only that there exists one such function making the Dirichlet integral finite. This concept is named after the German mathematician Lejeune Dirichlet.

Since the Dirichlet integral is nonnegative, 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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