Dirichlet energy

• Dirichlet's energy

Named after German mathematician Peter Gustav Lejeune Dirichlet (1805-1859), who made significant contributions to the theory of Fourier series.

Dirichlet energy (plural Dirichlet energies)

1. (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.
   • 2005, Roger Moser, Partial Regularity for Harmonic Maps and Related Problems, World Scientific, page 1:

     Variational principles play an important role in both geometry and physics, and one of the key problems with applications in both fields is the variational problem associated to the Dirichlet energy of maps between Riemannian manifolds.

   • 2011, Camillo De Lellis, Emanuele Nunzio Spadaro, Q-valued Functions Revisited, American Mathematical Society, page 28:

     The Dirichlet energy of a function ${\displaystyle f\in W^{1,2}}$ can be recovered, moreover, as the energy of the composition ${\displaystyle \xi _{BW}\circ f}$, where ${\displaystyle \xi _{BW}}$ is the biLipschitz embedding in Corollary 2.2 (compare with Theorem 2.4).

   • 2015, Sören Bartels, Numerical Methods for Nonlinear Partial Differential Equations, Springer, page 89:

     Since the Dirichlet energy is weakly lower semicontinuous and strongly continuous, the linear lower-order terms are weakly continuous on ${\displaystyle H_{D}^{1}(\Omega )}$, and since the finite element spaces are dense in ${\displaystyle H_{D}^{1}(\Omega )}$, we verify that ${\displaystyle I_{h}\to ^{\Gamma }I}$ as ${\displaystyle h\to 0}$.

• In mathematical terms, given an open set ${\displaystyle \Omega \subseteq \mathbb {R} ^{n}}$ and a function ${\displaystyle u:\Omega \to \mathbb {R} }$, the Dirichlet energy of ${\displaystyle u}$ is ${\displaystyle \textstyle E[u]={\frac {1}{2}}\int _{\Omega }\|\nabla u(x)\|^{2}\,dx}$, where ${\displaystyle \nabla u:\Omega \to \mathbb {R} ^{n}}$ denotes the gradient vector field of ${\displaystyle u}$.

• Calculus of variations on Wikipedia.Wikipedia
• Dirichlet's principle on Wikipedia.Wikipedia
• Harmonic map on Wikipedia.Wikipedia
• Laplace's equation on Wikipedia.Wikipedia