Dirichlet energy

plural Dirichlet energies · alternative Dirichlet's energy

1. functional analysis; Fourier analysis; functional analysis; Fourier analysismathematical analysisA 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.

   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

   2 more examples

   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

   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