Laplace Equation and Laplacian Operator |
In General > s.a. spectral geometry [quantum billiard].
$ Def: The partial differential equation
∇2 u:= gab ∇a ∇b 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.
@ Related topics:
Majic & Le Ru a1907 [new class of solutions].
> 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|−1/2 ∂i(|g|1/2 gij ∂j 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 [S\(^3\)];
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 PA(19)-a1708 [self-adjointness of sub-Laplacians];
Greenblatt a2102 [on polygonal domains].
> 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 JPA(18)-a1706 [and some solutions].
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