View source for Dirichlet's principle

{{Short description|Concept in potential theory}}
{{distinguish|Pigeonhole principle}}
In [[mathematics]], and particularly in [[potential theory]], '''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 [[Poisson's equation]] 

:<math>\Delta u + f = 0</math>

on a [[domain of a function|domain]] <math>\Omega</math> of <math>\mathbb{R}^n</math> with [[boundary condition]]

:<math>u=g</math> on the [[Boundary (topology) | boundary]] <math>\partial\Omega</math>,

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

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

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 [[Peter Gustav Lejeune Dirichlet]].

==History==
The name "Dirichlet's principle" is due to [[Bernhard Riemann]], who applied it in the study of [[Complex analytic function | complex analytic functions]].<ref>Monna 1975, p. 33</ref>

Riemann (and others such as [[Carl Friedrich Gauss]] and [[Peter Gustav Lejeune Dirichlet]]) knew that Dirichlet's integral is bounded below, which establishes the existence of an [[infimum]]; however, he took for granted the existence of a function that attains the minimum. [[Karl Weierstrass]] published the first criticism of this assumption in 1870, giving an example of a functional that has a greatest lower bound which is not a minimum value. Weierstrass's example was the functional

:<math>J(\varphi) = \int_{-1}^{1} \left( x \frac{d\varphi}{dx} \right)^2 \, dx </math>

where <math>\varphi</math> is continuous on <math>[-1,1]</math>, continuously differentiable on <math>(-1,1)</math>, and subject to boundary conditions <math>\varphi(-1)=a</math>, <math>\varphi(1)=b</math> where <math>a</math> and <math>b</math> are constants and <math>a \ne b</math>. Weierstrass showed that <math>\textstyle \inf_\varphi J(\varphi) = 0</math>, but no admissible function <math>\varphi</math> can make <math>J(\varphi)</math> equal 0. This example did not disprove Dirichlet's principle ''per se'', since the example integral is different from Dirichlet's integral. But it did undermine the reasoning that Riemann had used, and spurred interest in proving Dirichlet's principle as well as broader advancements in the [[calculus of variations]] and ultimately [[functional analysis]].<ref>Monna 1975, p. 33–37,43–44</ref><ref>Giaquinta and Hildebrand, p. 43–44</ref>

In 1900, [[David Hilbert|Hilbert]] later justified Riemann's use of Dirichlet's principle by developing the [[direct method in the calculus of variations]].<ref>Monna 1975, p. 55–56, citing {{citation | last=Hilbert | first=David | title=Über das Dirichletsche Prinzip | journal=Journal für die reine und angewandte Mathematik | year=1905 | volume=1905 | issue=129 | pages=63–67 | doi=10.1515/crll.1905.129.63 | s2cid=120074769 | language=de}}</ref>

==See also==
* [[Dirichlet problem]]
* [[Hilbert's twentieth problem]]
* [[Plateau's problem]]
* [[Green's identities#Green's first identity|Green's first identity]]

==Notes==
{{reflist}}

==References==
*{{citation|last=Courant|first= R.|author-link=Richard Courant|title=Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces. Appendix by M. Schiffer|publisher= Interscience |year= 1950}}
* {{citation | author=Lawrence C. Evans | title=Partial Differential Equations |author-link=Lawrence C. Evans | publisher=American Mathematical Society | year=1998 | isbn=978-0-8218-0772-9 }}
* {{citation | last1=Giaquinta | first1=Mariano | author1-link=Mariano Giaquinta | last2=Hildebrandt | first2=Stefan | title=Calculus of Variations I | publisher=Springer | year=1996 }}
* {{citation | author=A. F. Monna | title=Dirichlet's principle: A mathematical comedy of errors and its influence on the development of analysis | publisher=Oosthoek, Scheltema & Holkema | year=1975 }}
* {{MathWorld | urlname=DirichletsPrinciple | title=Dirichlet's Principle}}

[[Category:Calculus of variations]]
[[Category:Partial differential equations]]
[[Category:Harmonic functions]]
[[Category:Mathematical principles]]