Dictionary

Dirichlet energy en · NOUN

Etymology

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

Meanings

1. (qualifier:functional analysis; Fourier 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.
   • Since the Dirichlet energy is weakly lower semicontinuous and strongly continuous, the linear lower-order terms are weakly continuous on H#95;D¹(#92;Omega), and since the finite element spaces are dense in H#95;D¹(#92;Omega), we verify that I#95;h#92;to#92;GammaI as h#92;to 0. 2015, Sören Bartels, Numerical Methods for Nonlinear Partial Differential Equations, Springer, page 89:
   • 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. 2005, Roger Moser, Partial Regularity for Harmonic Maps and Related Problems, World Scientific, page 1:
   • The Dirichlet energy of a function f#92;inW#123;1,2#125; can be recovered, moreover, as the energy of the composition #92;xi#95;#123;BW#125;#92;circf, where #92;xi#95;#123;BW#125; is the biLipschitz embedding in Corollary 2.2 (compare with Theorem 2.4). 2011, Camillo De Lellis, Emanuele Nunzio Spadaro, Q-valued Functions Revisited, American Mathematical Society, page 28:

Forms

Translations (4)

nl Dirichlet-energie (functional that maps a function to a real number representing its variability) · bg енергия на Дирихле (functional that maps a function to a real number representing its variability) · it energia di Dirichlet (functional that maps a function to a real number representing its variability) · es energía de Dirichlet (functional that maps a function to a real number representing its variability)