"Dirichlet energy" meaning in English

{
  "etymology_text": "Named after German mathematician Peter Gustav Lejeune Dirichlet (1805-1859), who made significant contributions to the theory of Fourier series.",
  "forms": [
    {
      "form": "Dirichlet energies",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Dirichlet energy (plural Dirichlet energies)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
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          "kind": "other",
          "name": "English entries with incorrect language header",
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            "Entries with incorrect language header",
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        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Functional analysis",
          "orig": "en:Functional analysis",
          "parents": [
            "Mathematical analysis",
            "Mathematics",
            "Formal sciences",
            "Sciences",
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            "Fundamental"
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        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematical analysis",
          "orig": "en:Mathematical analysis",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "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.",
          "type": "quotation"
        },
        {
          "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).",
          "type": "quotation"
        },
        {
          "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.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "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."
      ],
      "id": "en-Dirichlet_energy-en-noun-nyQpQP1x",
      "links": [
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          "mathematical analysis",
          "mathematical analysis"
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        [
          "functional analysis",
          "functional analysis"
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        [
          "quadratic",
          "quadratic"
        ],
        [
          "functional",
          "functional"
        ],
        [
          "real function",
          "real function"
        ],
        [
          "real number",
          "real number"
        ],
        [
          "variable",
          "variable"
        ]
      ],
      "qualifier": "functional analysis; Fourier analysis; functional analysis; Fourier analysis",
      "raw_glosses": [
        "(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."
      ],
      "synonyms": [
        {
          "word": "Dirichlet's energy"
        }
      ],
      "topics": [
        "mathematical-analysis",
        "mathematics",
        "sciences"
      ],
      "translations": [
        {
          "code": "nl",
          "lang": "Dutch",
          "sense": "functional that maps a function to a real number representing its variability",
          "tags": [
            "feminine"
          ],
          "word": "Dirichlet-energie"
        },
        {
          "code": "it",
          "lang": "Italian",
          "sense": "functional that maps a function to a real number representing its variability",
          "tags": [
            "feminine"
          ],
          "word": "energia di Dirichlet"
        },
        {
          "code": "es",
          "lang": "Spanish",
          "sense": "functional that maps a function to a real number representing its variability",
          "tags": [
            "feminine"
          ],
          "word": "energía de Dirichlet"
        }
      ],
      "wikipedia": [
        "Dirichlet energy",
        "Peter Gustav Lejeune Dirichlet"
      ]
    }
  ],
  "word": "Dirichlet energy"
}

{
  "etymology_text": "Named after German mathematician Peter Gustav Lejeune Dirichlet (1805-1859), who made significant contributions to the theory of Fourier series.",
  "forms": [
    {
      "form": "Dirichlet energies",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Dirichlet energy (plural Dirichlet energies)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English entries with language name categories using raw markup",
        "English eponyms",
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        "English multiword terms",
        "English nouns",
        "English terms with non-redundant non-automated sortkeys",
        "English terms with quotations",
        "en:Functional analysis",
        "en:Mathematical analysis"
      ],
      "examples": [
        {
          "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.",
          "type": "quotation"
        },
        {
          "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).",
          "type": "quotation"
        },
        {
          "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.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "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."
      ],
      "links": [
        [
          "mathematical analysis",
          "mathematical analysis"
        ],
        [
          "functional analysis",
          "functional analysis"
        ],
        [
          "quadratic",
          "quadratic"
        ],
        [
          "functional",
          "functional"
        ],
        [
          "real function",
          "real function"
        ],
        [
          "real number",
          "real number"
        ],
        [
          "variable",
          "variable"
        ]
      ],
      "qualifier": "functional analysis; Fourier analysis; functional analysis; Fourier analysis",
      "raw_glosses": [
        "(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."
      ],
      "topics": [
        "mathematical-analysis",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "Dirichlet energy",
        "Peter Gustav Lejeune Dirichlet"
      ]
    }
  ],
  "synonyms": [
    {
      "word": "Dirichlet's energy"
    }
  ],
  "translations": [
    {
      "code": "nl",
      "lang": "Dutch",
      "sense": "functional that maps a function to a real number representing its variability",
      "tags": [
        "feminine"
      ],
      "word": "Dirichlet-energie"
    },
    {
      "code": "it",
      "lang": "Italian",
      "sense": "functional that maps a function to a real number representing its variability",
      "tags": [
        "feminine"
      ],
      "word": "energia di Dirichlet"
    },
    {
      "code": "es",
      "lang": "Spanish",
      "sense": "functional that maps a function to a real number representing its variability",
      "tags": [
        "feminine"
      ],
      "word": "energía de Dirichlet"
    }
  ],
  "word": "Dirichlet energy"
}