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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.
Dirichlet's principle states that, if the function u ( x ) is the solution to Poisson's equation
on a domain \Omega of \mathbb{R}^n with boundary condition
then u can be obtained as the minimizer of the Dirichlet's energy
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.
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.
Logic, Set theory, Statistics, Number theory, Mathematical logic
Mathematics, Mathematical logic, Germany, University of Königsberg, Hilbert space
Peter Gustav Lejeune Dirichlet, Dirichlet's theorem on arithmetic progressions, Dirichlet's approximation theorem, Dirichlet's unit theorem, Dirichlet conditions
Mathematics, Potential theory, Maximum principle, Electrostatics, David Hilbert