Laplace operator

English

Noun

1. Template:Lb A differential operator,denoted ∆ and defined on ${ℝ}^n$ as $\mathrm{\Delta }=\sum _{i=1}^n\frac{{\partial }^2}{\partial x_i^2}$, used in the modeling of wave propagation, heat flow and many other applications.
   • 1975, Various translators, V. Ja. Sikjrjavyĭ, A Quasidifferentiation Operator and Boundary Value Problems Connected With It, V. I. Averbvh, M. S. Birman, A. A. Blahin (editors), Transactions of the Moscow Mathematical Society for the Year 1972, Volume 27, [ТРУДЫ МОСКОВКОГО МАТЕМАҬИЧЕСКОГО ОБЩЕСТВА ТОМ 27 (1972)], American Mathematical Society, page 202,

     The first notion of a Laplace operator for functionals on a Hilbert space was introduced by Levy in [l], and the idea was developed further in [2]. Levy's results depended on the posthumous work of Gateaux [3] in which the Dirichlet problem in Hilbert space was considered (without any concise definition of the Laplace operator).

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Usage notes

May be regarded as the divergence (∇·) of the gradient (∇) of a function; i.e. Δ = ∇·∇ (= ∇²).

The class of elliptic operators is a generalization.

Synonyms

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Hypernyms

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