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A partial differential equation which is satisfied by the [[Potential|potential]] of a mass distribution inside domains occupied by the masses creating this potential. For the [[Newton potential|Newton potential]] in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073290/p0732901.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073290/p0732902.png" />, and the [[Logarithmic potential|logarithmic potential]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073290/p0732903.png" /> the Poisson equation has the form
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A partial differential equation which is satisfied by the [[Potential|potential]] of a mass distribution inside domains occupied by the masses creating this potential. For the [[Newton potential|Newton potential]] in the space $\mathbf R^n$, $n\geq3$, and the [[Logarithmic potential|logarithmic potential]] in $\mathbf R^3$ the Poisson equation has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073290/p0732904.png" /></td> </tr></table>
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$$\Delta u=\sum_{i=1}^n\frac{\partial^2u}{\partial x_i^2}=-\sigma(S^n)\rho(x_1,\dots,x_n),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073290/p0732905.png" /> is the density of the mass distribution, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073290/p0732906.png" /> is the area of the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073290/p0732907.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073290/p0732908.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073290/p0732909.png" /> is the value of the gamma-function.
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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).
 
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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,  "Equations of mathematical physics" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,  "Equations of mathematical physics" , MIR  (1980)  (Translated from Russian) {{MR|0587310}} {{MR|0581247}} {{ZBL|0499.35002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German) {{MR|0195654}} {{ZBL|}} </TD></TR></table>
  
  
  
 
====Comments====
 
====Comments====
The map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073290/p07329010.png" /> defines a morphism from the sheaf of local differences of superharmonic functions into a sheaf of measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073290/p07329011.png" />. This remark leads to a treatment of the Poisson problem in the framework of harmonic spaces (cf. [[Harmonic space|Harmonic space]]), see [[#References|[a1]]].
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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|Harmonic space]]), see [[#References|[a1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.-Y. Maeda,  "Dirichlet integrals on harmonic spaces" , ''Lect. notes in math.'' , '''803''' , Springer  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Rudin,  "Function theory in the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073290/p07329012.png" />" , Springer  (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  O.D. Kellogg,  "Foundations of potential theory" , F. Ungar  (1929)  (Re-issue: Springer, 1967)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.-Y. Maeda,  "Dirichlet integrals on harmonic spaces" , ''Lect. notes in math.'' , '''803''' , Springer  (1980) {{MR|0576059}} {{ZBL|0426.31001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Rudin,  "Function theory in the unit ball in $\mathbf C^n$" , Springer  (1980) {{MR|601594}} {{ZBL|0495.32001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  O.D. Kellogg,  "Foundations of potential theory" , F. Ungar  (1929)  (Re-issue: Springer, 1967) {{MR|0222317}} {{MR|1522134}} {{ZBL|0152.31301}} {{ZBL|0053.07301}} </TD></TR></table>

Latest revision as of 06:54, 27 August 2014

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
How to Cite This Entry:
Poisson equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_equation&oldid=14096
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article