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''in the broad sense''
 
''in the broad sense''
  
A [[Potential|potential]] with Newton kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n0665801.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n0665802.png" /> is the distance between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n0665803.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n0665804.png" /> of the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n0665805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n0665806.png" />, that is, an integral of the form
+
A [[Potential|potential]] with Newton kernel $1/|x-y|^{N-2}$, where $|x-y|$ is the distance between two points $x$ and $y$ of the Euclidean space $\mathbf R^N$, $N\geq3$, that is, an integral of 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/n/n066/n066580/n0665807.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$u(x)=\int\limits_S\frac{d\mu(y)}{|x-y|^{N-2}},\tag{1}$$
  
where integration is with respect to a certain [[Radon measure|Radon measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n0665808.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n0665809.png" /> with compact support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658010.png" />. When the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658011.png" /> is non-negative, the Newton potential (1) is a superharmonic function in the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658012.png" /> (see [[Subharmonic function|Subharmonic function]]).
+
where integration is with respect to a certain [[Radon measure|Radon measure]] $\mu$ on $\mathbf R^N$ with compact support $S$. When the measure $\mu$ is non-negative, the Newton potential \ref{1} is a superharmonic function in the whole space $\mathbf R^n$ (see [[Subharmonic function|Subharmonic function]]).
  
Outside the support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658014.png" /> the Newton potential (1) has derivatives of all orders in the coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658015.png" /> and is a regular solution of the [[Laplace equation|Laplace equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658016.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658017.png" /> is a [[Harmonic function|harmonic function]] on the open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658018.png" /> and is regular at infinity with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658019.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658020.png" /> is absolutely continuous, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658021.png" /> has the form
+
Outside the support $S$ of $\mu$ the Newton potential \ref{1} has derivatives of all orders in the coordinates of $x$ and is a regular solution of the [[Laplace equation|Laplace equation]] $\Delta u=0$, that is, $u$ is a [[Harmonic function|harmonic function]] on the open set $CS$ and is regular at infinity with $u(\infty)=0$. When $\mu$ is absolutely continuous, then $u$ 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/n/n066/n066580/n06658022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$u(x)=\int\limits_D\frac{1}{|x-y|^{N-2}}f(y)d\omega(y),\tag{2}$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658023.png" /> is the volume element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658025.png" /> is a certain bounded domain. If here the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658026.png" /> is Hölder continuous in the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658027.png" /> and if the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658028.png" /> consists of finitely many closed Lyapunov hypersurfaces (cf. [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658029.png" /> has continuous second-order derivatives inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658030.png" /> and satisfies the [[Poisson equation|Poisson equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658031.png" />.
+
where $d\omega$ is the volume element in $\mathbf R^N$ and $D$ is a certain bounded domain. If here the density $f$ is Hölder continuous in the closed domain $D$ and if the boundary $\partial D$ consists of finitely many closed Lyapunov hypersurfaces (cf. [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]), then $u$ has continuous second-order derivatives inside $D$ and satisfies the [[Poisson equation|Poisson equation]] $\Delta u(x)=-(N-2)2\pi^{N/2}f(x)/\Gamma(N/2)$.
  
In Newton's work the concept of a  "potential"  does not yet occur. The existence of a force function for Newtonian gravitational forces was first proved by J.L. Lagrange in 1773. The terms  "potential function"  and  "potential"  applied to integrals of the form (2) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658032.png" /> were first used by G. Green in 1828 and C.F. Gauss in 1840. The term  "Newton potential"  is sometimes used in the narrow sense, applied only to volume potentials of the form (2), and sometimes only to the physically real case of a potential (2) of gravitational forces for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658033.png" />, created by masses distributed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658034.png" /> with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658035.png" />.
+
In Newton's work the concept of a  "potential"  does not yet occur. The existence of a force function for Newtonian gravitational forces was first proved by J.L. Lagrange in 1773. The terms  "potential function"  and  "potential"  applied to integrals of the form \ref{2} for $N=3$ were first used by G. Green in 1828 and C.F. Gauss in 1840. The term  "Newton potential"  is sometimes used in the narrow sense, applied only to volume potentials of the form \ref{2}, and sometimes only to the physically real case of a potential \ref{2} of gravitational forces for $N=3$, created by masses distributed in $D$ with density $f(y)$.
  
If an integral of type (2) or (1) is over a hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658036.png" />, that is, if
+
If an integral of type \ref{2} or \ref{1} is over a hypersurface $S\subset\mathbf R^N$, that is, if
  
<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/n/n066/n066580/n06658037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$u(x)=\int\limits_S\frac{1}{|x-y|^{N-2}}f(y)d\sigma(y),\tag{3}$$
  
then one speaks of a simple-layer Newton potential; it is a regular harmonic function everywhere outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658038.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658039.png" /> is a closed Lyapunov hypersurface and the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658040.png" /> is Hölder continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658041.png" />, then the simple-layer Newton potential is continuous everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658042.png" />, and its derivatives are continuous outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658043.png" />. Moreover, its normal derivative in the direction of the outward normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658044.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658045.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658046.png" /> has different limits on approaching <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658047.png" /> from the inside and the outside. These are expressed by the formulas
+
then one speaks of a simple-layer Newton potential; it is a regular harmonic function everywhere outside $S$. If $S$ is a closed Lyapunov hypersurface and the density $f(y)$ is Hölder continuous on $S$, then the simple-layer Newton potential is continuous everywhere on $\mathbf R^N$, and its derivatives are continuous outside $S$. Moreover, its normal derivative in the direction of the outward normal $n_0$ to $S$ at $y_0\in S$ has different limits on approaching $S$ from the inside and the outside. These are expressed by the formulas
  
<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/n/n066/n066580/n06658048.png" /></td> </tr></table>
+
$$\lim_{x\to y_0}\left.\frac{du}{dn_0}\right|_i=\frac{du(y_0)}{dn_0}+\frac{(N-2)\pi^{N/2}}{\Gamma(N/2)}f(y_0),$$
  
<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/n/n066/n066580/n06658049.png" /></td> </tr></table>
+
$$\lim_{x\to y_0}\left.\frac{du}{dn_0}\right|_o=\frac{du(y_0)}{dn_0}-\frac{(N-2)\pi^{N/2}}{\Gamma(N/2)}f(y_0),$$
  
 
where
 
where
  
<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/n/n066/n066580/n06658050.png" /></td> </tr></table>
+
$$\frac{du(y_0)}{dn_0}=(N-2)\int\limits_Sf(y)\frac{\cos(y-y_0,n_0)}{|y-y_0|^{N-1}}d\sigma(y),\quad y_0\in S,$$
  
is the so-called direct value of the normal derivative of the simple-layer Newton potential, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658051.png" /> is the angle between the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658052.png" /> and the normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658053.png" />; the normal derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658054.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658055.png" />.
+
is the so-called direct value of the normal derivative of the simple-layer Newton potential, and $(y-y_0,n_0)$ is the angle between the vector $y-y_0$ and the normal $n_0$; the normal derivative $du(y_0)/dn_0$ is continuous on $S$.
  
 
A double-layer Newton potential has the form
 
A double-layer Newton potential 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/n/n066/n066580/n06658056.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$v(x)=\int\limits_Sf(y)\frac{\cos(y-x,n)}{|y-x|^{N-1}}d\sigma y,\tag{4}$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658057.png" /> is the outward normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658058.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658059.png" />. It is also a harmonic function outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658060.png" />, but upon approaching <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658061.png" /> it has a discontinuity. Under the same assumptions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658063.png" /> it has limits from the inside and the outside of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658064.png" />. These are expressed by the formulas
+
where $n$ is the outward normal to $S$ at $y\in S$. It is also a harmonic function outside $S$, but upon approaching $S$ it has a discontinuity. Under the same assumptions on $S$ and $f(y)$ it has limits from the inside and the outside of $S$. These are expressed by the formulas
  
<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/n/n066/n066580/n06658065.png" /></td> </tr></table>
+
$$\lim_{x\to y_0}\left.v(x)\right|_i=v(y_0)+\frac{\pi^{N/2}}{\Gamma(N/2)}f(y_0),$$
  
<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/n/n066/n066580/n06658066.png" /></td> </tr></table>
+
$$\lim_{x\to y_0}\left.v(x)\right|_o=v(y_0)-\frac{\pi^{N/2}}{\Gamma(N/2)}f(y_0),$$
  
 
where
 
where
  
<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/n/n066/n066580/n06658067.png" /></td> </tr></table>
+
$$v(y_0)=\int\limits_Sf(y)\frac{\cos(y-y_0,n)}{|y-y_0|^{N-1}}d\sigma(y)$$
  
is the so-called direct value of the double-layer Newton potential at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658068.png" />. Under somewhat more stringent conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658070.png" /> the normal derivative of the double-layer Newton potential is, however, continuous on passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066580/n06658071.png" />.
+
is the so-called direct value of the double-layer Newton potential at $y_0\in S$. Under somewhat more stringent conditions on $S$ and $f(y)$ the normal derivative of the double-layer Newton potential is, however, continuous on passing through $S$.
  
 
See also [[Double-layer potential|Double-layer potential]]; [[Potential theory|Potential theory]]; [[Simple-layer potential|Simple-layer potential]]; [[Surface potential|Surface potential]].
 
See also [[Double-layer potential|Double-layer potential]]; [[Potential theory|Potential theory]]; [[Simple-layer potential|Simple-layer potential]]; [[Surface potential|Surface potential]].

Revision as of 15:02, 23 August 2014

in the broad sense

A potential with Newton kernel $1/|x-y|^{N-2}$, where $|x-y|$ is the distance between two points $x$ and $y$ of the Euclidean space $\mathbf R^N$, $N\geq3$, that is, an integral of the form

$$u(x)=\int\limits_S\frac{d\mu(y)}{|x-y|^{N-2}},\tag{1}$$

where integration is with respect to a certain Radon measure $\mu$ on $\mathbf R^N$ with compact support $S$. When the measure $\mu$ is non-negative, the Newton potential \ref{1} is a superharmonic function in the whole space $\mathbf R^n$ (see Subharmonic function).

Outside the support $S$ of $\mu$ the Newton potential \ref{1} has derivatives of all orders in the coordinates of $x$ and is a regular solution of the Laplace equation $\Delta u=0$, that is, $u$ is a harmonic function on the open set $CS$ and is regular at infinity with $u(\infty)=0$. When $\mu$ is absolutely continuous, then $u$ has the form

$$u(x)=\int\limits_D\frac{1}{|x-y|^{N-2}}f(y)d\omega(y),\tag{2}$$

where $d\omega$ is the volume element in $\mathbf R^N$ and $D$ is a certain bounded domain. If here the density $f$ is Hölder continuous in the closed domain $D$ and if the boundary $\partial D$ consists of finitely many closed Lyapunov hypersurfaces (cf. Lyapunov surfaces and curves), then $u$ has continuous second-order derivatives inside $D$ and satisfies the Poisson equation $\Delta u(x)=-(N-2)2\pi^{N/2}f(x)/\Gamma(N/2)$.

In Newton's work the concept of a "potential" does not yet occur. The existence of a force function for Newtonian gravitational forces was first proved by J.L. Lagrange in 1773. The terms "potential function" and "potential" applied to integrals of the form \ref{2} for $N=3$ were first used by G. Green in 1828 and C.F. Gauss in 1840. The term "Newton potential" is sometimes used in the narrow sense, applied only to volume potentials of the form \ref{2}, and sometimes only to the physically real case of a potential \ref{2} of gravitational forces for $N=3$, created by masses distributed in $D$ with density $f(y)$.

If an integral of type \ref{2} or \ref{1} is over a hypersurface $S\subset\mathbf R^N$, that is, if

$$u(x)=\int\limits_S\frac{1}{|x-y|^{N-2}}f(y)d\sigma(y),\tag{3}$$

then one speaks of a simple-layer Newton potential; it is a regular harmonic function everywhere outside $S$. If $S$ is a closed Lyapunov hypersurface and the density $f(y)$ is Hölder continuous on $S$, then the simple-layer Newton potential is continuous everywhere on $\mathbf R^N$, and its derivatives are continuous outside $S$. Moreover, its normal derivative in the direction of the outward normal $n_0$ to $S$ at $y_0\in S$ has different limits on approaching $S$ from the inside and the outside. These are expressed by the formulas

$$\lim_{x\to y_0}\left.\frac{du}{dn_0}\right|_i=\frac{du(y_0)}{dn_0}+\frac{(N-2)\pi^{N/2}}{\Gamma(N/2)}f(y_0),$$

$$\lim_{x\to y_0}\left.\frac{du}{dn_0}\right|_o=\frac{du(y_0)}{dn_0}-\frac{(N-2)\pi^{N/2}}{\Gamma(N/2)}f(y_0),$$

where

$$\frac{du(y_0)}{dn_0}=(N-2)\int\limits_Sf(y)\frac{\cos(y-y_0,n_0)}{|y-y_0|^{N-1}}d\sigma(y),\quad y_0\in S,$$

is the so-called direct value of the normal derivative of the simple-layer Newton potential, and $(y-y_0,n_0)$ is the angle between the vector $y-y_0$ and the normal $n_0$; the normal derivative $du(y_0)/dn_0$ is continuous on $S$.

A double-layer Newton potential has the form

$$v(x)=\int\limits_Sf(y)\frac{\cos(y-x,n)}{|y-x|^{N-1}}d\sigma y,\tag{4}$$

where $n$ is the outward normal to $S$ at $y\in S$. It is also a harmonic function outside $S$, but upon approaching $S$ it has a discontinuity. Under the same assumptions on $S$ and $f(y)$ it has limits from the inside and the outside of $S$. These are expressed by the formulas

$$\lim_{x\to y_0}\left.v(x)\right|_i=v(y_0)+\frac{\pi^{N/2}}{\Gamma(N/2)}f(y_0),$$

$$\lim_{x\to y_0}\left.v(x)\right|_o=v(y_0)-\frac{\pi^{N/2}}{\Gamma(N/2)}f(y_0),$$

where

$$v(y_0)=\int\limits_Sf(y)\frac{\cos(y-y_0,n)}{|y-y_0|^{N-1}}d\sigma(y)$$

is the so-called direct value of the double-layer Newton potential at $y_0\in S$. Under somewhat more stringent conditions on $S$ and $f(y)$ the normal derivative of the double-layer Newton potential is, however, continuous on passing through $S$.

See also Double-layer potential; Potential theory; Simple-layer potential; Surface potential.

References

[1] N.M. Günter, "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar (1967) (Translated from French)
[2] L.N. Sretenskii, "Theory of the Newton potential" , Moscow-Leningrad (1946) (In Russian)
[3] N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)
[4] M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)
[5] J. Wermer, "Potential theory" , Lect. notes in math. , 408 , Springer (1974)
[6] O.D. Kellogg, "Foundations of potential theory" , F. Ungar (1929) (Re-issue: Springer, 1967)


Comments

The "analogue" in dimension 2 is the logarithmic potential.

References

[a1] C.F. Gauss, "Allgemeine Lehrsätze in Beziehung auf die im verkehrte Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungskräfte" , Werke , 5 , K. Gesellschaft Wissenschaft. Göttingen (1876) pp. 195–242
[a2a] G. Green, "An essay on the application of mathematical analysis to the theories of electricity and magnetism I" J. Reine Angew. Math. , 39 (1850) pp. 73–89 (Re-issued by Lord Kelvin)
[a2b] G. Green, "An essay on the application of mathematical analysis to the theories of electricity and magnetism II" J. Reine Angew. Math. , 44 (1852) pp. 356–374 (Re-issued by Lord Kelvin)
[a2c] G. Green, "An essay on the application of mathematical analysis to the theories of electricity and magnetism III" J. Reine Angew. Math. , 47 (1854) pp. 161–221 (Re-issued by Lord Kelvin)
[a3] J.-L. Lagrange, "Sur l'équation séculaire de la lune" Mém. Acad. Roy. Sci. Paris (1773)
How to Cite This Entry:
Newton potential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Newton_potential&oldid=33114
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article