Dirichlet energy

== English == === Alternative forms === Dirichlet's energy === Etymology === Named after German mathematician Peter Gustav Lejeune Dirichlet (1805–1859), who made significant contributions to the theory of Fourier series. === Noun === Dirichlet energy (plural Dirichlet energies) (mathematical analysis, functional analysis, Fourier analysis) A quadratic functional which, given a real function defined on an open subset of ℝn, yields a real number that is a measure of how variable said function is. ==== Usage notes ==== In mathematical terms, given an open set Ω ⊆ R n {\displaystyle \Omega \subseteq \mathbb {R} ^{n}} and a function u : Ω → R {\displaystyle u:\Omega \to \mathbb {R} } , the Dirichlet energy of u {\displaystyle u} is E [ u ] = 1 2 ∫ Ω ‖ ∇ u ( x ) ‖ 2 d x {\displaystyle \textstyle E[u]={\frac {1}{2}}\int _{\Omega }\|\nabla u(x)\|^{2}\,dx} , where ∇ u : Ω → R n {\displaystyle \nabla u:\Omega \to \mathbb {R} ^{n}} denotes the gradient vector field of u {\displaystyle u} . ==== Translations ==== === Further reading === Calculus of variations on Wikipedia.Wikipedia Dirichlet's principle on Wikipedia.Wikipedia Harmonic map on Wikipedia.Wikipedia Laplace's equation on Wikipedia.Wikipedia