Dirichlet's principle

Not to be confused with Pigeonhole principle.

In mathematics, Dirichlet's principle in potential theory states that, if the function u(x) is the solution to Poisson's equation

$\Delta u %2B f = 0\,$

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

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

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

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

amongst all twice differentiable functions $v$ such that $v=g$ on $\partial\Omega$ (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.

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" from MathWorld.

Dirichlet's principle

See also

References