Poisson equation
A partial differential equation which is satisfied by the
potential
of a mass distribution inside domains occupied by
the masses creating this potential. For the
Newton potential
in the space
 ,
 ,
and the
logarithmic potential
in
the Poisson equation has the form
where
is the density of the mass distribution,
is the area of the unit sphere
in
and
is the value of the gamma-function.
Poisson's equation is a basic example of a non-homogeneous equation of
elliptic type. The equation was first considered by
S. Poisson
(1812).
References| [1] |
A.V. Bitsadze,
"Equations of mathematical physics"
, MIR
(1980)
(Translated from Russian) | | [2] |
R. Courant,
D. Hilbert,
"Methods of mathematical physics. Partial differential equations"
, 2
, Interscience
(1965)
(Translated from German) |
E.D. Solomentsev
CommentsThe map
defines a morphism from the sheaf of local differences
of superharmonic functions into a sheaf of measures on
 .
This remark leads to a treatment of the Poisson
problem in the framework of harmonic spaces (cf.
Harmonic space),
see
[a1].
References| [a1] |
F.-Y. Maeda,
"Dirichlet integrals on harmonic spaces"
, Lect. notes in math.
, 803
, Springer
(1980) | | [a2] |
S.D. Poisson,
"Remarques sur une équation qui se présente dans la théorie des attractions des sphéroïdes"
Nouveau Bull. Soc. Philomathique de Paris
, 3
(1813)
pp. 388–392 | | [a3] |
W. Rudin,
"Function theory in the unit ball in
 "
, Springer
(1980) | | [a4] |
O.D. Kellogg,
"Foundations of potential theory"
, F. Ungar
(1929)
(Re-issue: Springer, 1967) |
This text originally appeared in Encyclopaedia of Mathematics
- ISBN 1402006098
|