Misplaced Pages

This is an old revision of this page, as edited by Peak (talk | contribs) at 04:08, 21 February 2005 (==See also == (The Pigeonhole Principle is another "Dirichlet principle")). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

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

${\displaystyle u=g}$ on ${\displaystyle \partial \Omega }$

can be obtained as the minimizer of the Dirichlet integral

${\displaystyle \int _{\Omega }|\nabla v|^{2}}$

amongst all functions

${\displaystyle v}$ such that ${\displaystyle v=g}$ on ${\displaystyle \partial \Omega }$,

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.

See also

• Pigeonhole principle (a principle in combinatorics)

• Mathematical analysis
• Calculus of variations
• Partial differential equations
• Potential theory