Laplace operator

Noun

Laplace operator (plural Laplace operators)

1. A differential operator, denoted ∆ and defined on ℝⁿ as Δ=∑ᵢ₌₁ⁿ(∂²)/(∂x_i²), used in the modeling of wave propagation, heat flow and many other applications.
   • 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...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...
   • 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).2006, Vadim Kostrykin, Robert Schrader, “Laplacians on Metric Graphs: Eigenvalues, Resolvents, and Semigroups”, in Gregory Berkolaiko, Robert Carlson, Stephen A. Fulling, Peter Kuchment, editors...

   Synonyms:

   Laplacian