Laplace
Equation| Laplace
Equation |
In General > s.a. spectral
geometry [quantum billiard].
$ Def: The 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,
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 R2];
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]; > s.a. green function, Neumann
Problem.
Laplacian (Laplace-Beltrami) Operator > s.a. Boundary
Value Problems; D'Alambertian.
$ Def: On forms, if d
is the exterior derivative, and 
=
(–1)p (*)–1 d
(*),
the operator
:=
–(d + d)
.
* 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.
@ On S2: Dennis JPA(04), JPA(05)mp/04 [Maxwell's
multipoles].
@ Discrete: Lee in(83) [simplicial complex]; Zakrzewski JNMP(05)ht/04 [lattice]; > s.a.
electricity [network]; graphs and
invariants.
@ 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 second eigenvalue]; Ammann & Humbert IJGMP(06)
[first eigenvalue]; Hu JMP(08)-a0805 [on
homogeneous spaces]; Ho DG&A(08) [first eigenvalue, bound from curvature].
@ Related topics: Ryan & Turbiner PLA(04)qp [conformal
invariance and factor ordering].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
30 jun 2008