Meaning of Dirichlet energy | Babel Free

Definitions

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.

Equivalents

Examples

“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.”

“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).”

“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.”

CEFR level