# Laplace operator

## English

### Noun

Laplace operator (plural Laplace operators)

1. () A differential operator,denoted and defined on ${\displaystyle \mathbb {R} ^{n}}$ as ${\displaystyle \Delta =\sum _{i=1}^{n}{\frac {\partial ^{2}}{\partial x_{i}^{2}}}}$, 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).
• 2006, Vadim Kostrykin, Robert Schrader, Laplacians on Metric Graphs: Eigenvalues, Resolvents, and Semigroups, Gregory Berkolaiko, Robert Carlson, Stephen A. Fulling, Peter Kuchment (editors), Quantum Graphs and Their Applications: Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference, American Mathematical Society, page 201,
Nevertheless, a systematic analysis of heat semigroups for Laplace operators on metric graphs is still missing. [] Below we will prove an upper bound on the number of negative eigenvalues (Theorem 3.7) and a lower bound on the spectrum of Laplace operators (Theorem 3.10).
• 2008, Shantanu Das, Functional Fractional Calculus for System Identification and Controls, Springer, page 153,
There are lots of approaches to get rational approximation of fractional order Laplace operator. [] The rational function approximation gives direction to realize this fractional order Laplace operator in circuit impedance and admittance forms. This section gives insight into simple method of approximating the fractional Laplace operator.

#### 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.