See Dirichlet energy in All languages combined, or Wiktionary
{
"etymology_text": "Named after German mathematician Peter Gustav Lejeune Dirichlet (1805-1859), who made significant contributions to the theory of Fourier series.",
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"ref": "2005, Roger Moser, Partial Regularity for Harmonic Maps and Related Problems, World Scientific, page 1:",
"text": "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.",
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{
"ref": "2011, Camillo De Lellis, Emanuele Nunzio Spadaro, Q-valued Functions Revisited, American Mathematical Society, page 28:",
"text": "The Dirichlet energy of a function f#x5C;inW#x7B;1,2#x7D; can be recovered, moreover, as the energy of the composition #x5C;xi#x5F;#x7B;BW#x7D;#x5C;circf, where #x5C;xi#x5F;#x7B;BW#x7D; is the biLipschitz embedding in Corollary 2.2 (compare with Theorem 2.4).",
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"ref": "2015, Sören Bartels, Numerical Methods for Nonlinear Partial Differential Equations, Springer, page 89:",
"text": "Since the Dirichlet energy is weakly lower semicontinuous and strongly continuous, the linear lower-order terms are weakly continuous on H#x5F;D¹(#x5C;Omega), and since the finite element spaces are dense in H#x5F;D¹(#x5C;Omega), we verify that I#x5F;h#x5C;to#x5C;GammaI as h#x5C;to 0.",
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"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."
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"(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."
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"ref": "2011, Camillo De Lellis, Emanuele Nunzio Spadaro, Q-valued Functions Revisited, American Mathematical Society, page 28:",
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This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-11-06 from the enwiktionary dump dated 2024-10-02 using wiktextract (fbeafe8 and 7f03c9b). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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