Laplace Equation and Laplacian Operator  

In General > s.a. spectral geometry [quantum billiard].
$ Def: The partial differential equation

2 u:= gabab u = 0 .

* Applications: Satisfied by the electrostatic potential in the absence of sources.
* Solution methods: Separation of variables, possible in 11 (known) coordinate systems [@ Morse & Feschbach 53, pp 509 & 655]; Holomorphic functions [@ in Panofsky & Phillips 62].
* Relationships: A special case of Poisson equation, whose solutions are called harmonic or potential functions.
@ Boundary-value problems: Minotti & Moreno JMP(90) [regions of \(\mathbb R\)2]; Esposito NCB(99)ht/98, err NCB(00) [for square (∇2)2]; Chechkin & Gadyl'shin mp/03 [perforated boundaries]; Gibou & Fedkiw JCP(05) [Dirichlet boundary conditions, 4th-order discretization]; Tatari & Dehghan PS(05) [disk, Adomian decomposition method]; Yaseen et al a1208 [DJ iterative method for exact solution]; > s.a. green function; Neumann Problem.
> In physics: see scalar field theories; klein-gordon fields.
> Online resources: see MathWorld page; Wikipedia page.

Laplacian (Laplace-Beltrami) Operator > s.a. Boundary-Value Problems; D'Alembertian.
$ Def: On forms, if d is the exterior derivative, and δ = (–1)p (*)–1 d (*), the operator

\(\square\):= –(δd + dδ) .

* For arbitrary coordinates: A useful expression is

2 f = |g|–1/2i(|g|1/2 gijj f) .

* On S2: The eigenvalues are l (l+1), each with a 2l + 1 degeneracy; Alternatively, any eigenfunction is given in terms of a unique set of directions, Maxwell's multipoles, whose existence and uniqueness is known as Sylvester's theorem; > s.a. spherical harmonics.
* On S3: The eigenvalues are –k (k + 2), each with a (k + 1)2 degeneracy.
@ General references: Styer AJP(15)dec [geometrical significance of the Laplacian, and wave equation for a drumhead].
@ On S2: Dennis JPA(04), JPA(05)mp/04 [Maxwell's multipoles].
@ Spectrum: Ozawa CMP(84) [on bounded domain / random set of balls]; Cornish & Turok CQG(98)gq [compact manifolds]; Lehoucq et al CQG(02)gq [3D spherical spaces]; Takahashi JGP(02) [and connected sums]; Post mp/02, mp/02 [non-compact]; Lachièze-Rey JPA(04)m.SP/04 [S3]; Dowker CQG(04) [on lens spaces]; Lachièze-Rey & Caillerie CQG(05) [3D spherical spaces]; Benguria & Linde mp/05 [hyperbolic space, bound on 2nd eigenvalue]; Ammann & Humbert IJGMP(06) [first eigenvalue]; Hu JMP(08)-a0805 [on homogeneous spaces]; Ho DG&A(08) [1st eigenvalue, bound from curvature]; Munteanu JDG(09) [1st eigenvalue, on Kähler manifolds]; Cianchi & Maz'ya JDG(11) [non-compact Riemannian manifolds].
@ Related topics: Ryan & Turbiner PLA(04)qp [conformal invariance and factor ordering]; Fehér & Pusztai RPMP(08) [isometry-reduced, self-adjointness]; Asorey et al IJGMP(15)-a1510 [topology and geometry of self-adjoint and elliptic boundary conditions]; Franceschi et al a1708 [self-adjointness of sub-Laplacians].
> Online resources: see MathWorld page; Wikipedia page.

Generalizations and Similar Operators > s.a. electricity [network]; graphs and graph invariants.
@ Discretization / on lattices: Zakrzewski JNMP(05)ht/04; Thampi et al JCP(13) [with isotropic discretization error, from lattice hydrodynamics]; Sridhar a1501 [asymptotic determinant of the discrete Laplacian].
@ On other discrete structures: Lee in(85) [simplicial complexes]; Begué et al Frac(13)-a1201 [Sierpiński carpet]; Derfel et al JPA(12)-a1206 [fractals]; Calcagni et al CQG(13)-a1208 [cellular complexes]; Badanin et al a1301 [on periodic discrete graphs]; Thüringen MG13(15)-a1302 [abstract simplicial complexes, and expectation value of the heat kernel trace].
@ For higher-spin fields: De Bie et al a1501; Eelbode et al a1706 [and some solutions].

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