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 $\mathbf R^n$, $n\geq3$, 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