Misplaced Pages

Dirichlet's principle: Difference between revisions

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Browse history interactively← Previous editNext edit →Content deleted Content addedVisualWikitext
Revision as of 04:08, 21 February 2005 editPeak (talk | contribs)Extended confirmed users3,658 editsm ==See also == (The Pigeonhole Principle is another "Dirichlet principle")← Previous edit Revision as of 04:17, 8 June 2006 edit undoCj67 (talk | contribs)303 edits The pigeonhole principle is unrelated. If there might be some confusion, there can be a disambiguation, but it shouldn't be under ``see also".Next edit →
Line 15: Line 15:
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
attain its minimum. ] later justified Riemann's use of Dirichlet's principle. attain its minimum. ] later justified Riemann's use of Dirichlet's principle.

==See also==
* ] (a principle in ])


]]]] ]]]]

Revision as of 04:17, 8 June 2006

In mathematics, Dirichlet's principle in potential theory states that the harmonic function u {\displaystyle u} on a domain Ω {\displaystyle \Omega } with boundary condition

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

can be obtained as the minimizer of the Dirichlet integral

∫ Ω | ∇ v | 2 {\displaystyle \int _{\Omega }|\nabla v|^{2}}

amongst all 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.

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.

Categories: