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 , n\geq3, and the logarithmic potential in \mathbf R^3 the Poisson equation has the form
\Delta u=\sum_{i=1}^n\frac{\partial^2u}{\partial x_i^2}=-\sigma(S^n)\rho(x_1,\dots,x_n),
where \rho=\rho(x_1,\dots,x_n) is the density of the mass distribution, \sigma(S^n)=n\pi^{n/2}/\Gamma(n/2+1) is the area of the unit sphere S^n in \mathbf R^n and \Gamma(n/2+1) 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) MR0587310 MR0581247 Zbl 0499.35002 |
[2] | R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) MR0195654 |
Comments
The map u\mapsto\Delta(u) defines a morphism from the sheaf of local differences of superharmonic functions into a sheaf of measures on \mathbf R^n. 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) MR0576059 Zbl 0426.31001 |
[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 \mathbf C^n" , Springer (1980) MR601594 Zbl 0495.32001 |
[a4] | O.D. Kellogg, "Foundations of potential theory" , F. Ungar (1929) (Re-issue: Springer, 1967) MR0222317 MR1522134 Zbl 0152.31301 Zbl 0053.07301 |
Poisson equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_equation&oldid=33144