Misplaced Pages

This is an old revision of this page, as edited by ZéroBot (talk | contribs) at 18:58, 23 January 2012 (r2.7.1) (Robot: Adding zh:狄利克雷原理). 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, if the function u(x) is the solution to Poisson's equation

${\displaystyle \Delta u+f=0\,}$

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

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

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

${\displaystyle E=\int _{\Omega }\left({\frac {1}{2}}|\nabla v|^{2}-vf\right)\,\mathrm {d} x}$

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

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

• Plateau's problem
• Green's first identity

References

• Courant, R. (1950), Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces. Appendix by M. Schiffer, Interscience
• Lawrence C. Evans (1998), Partial Differential Equations, American Mathematical Society, ISBN 978-0821807729
• Weisstein, Eric W. "Dirichlet's Principle". MathWorld.

• Mathematical analysis
• Calculus of variations
• Partial differential equations
• Harmonic functions
• Mathematical principles