Dirichlet's 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 $u ( x )$ is the solution to Poisson's equation

\Delta u+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 Peter Gustav Lejeune Dirichlet.

History

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 978-0-8218-0772-9
Weisstein, Eric W. "Dirichlet's Principle". MathWorld.

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Dirichlet's principle

Formal statement

History

See also

References