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 is the solution to Poisson's equation

on a domain of with boundary condition

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

amongst all twice differentiable functions such that on (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.


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 by the direct method in the calculus of variations.

See also


  • 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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