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''equilibrium problem, electrostatic problem''
 
''equilibrium problem, electrostatic problem''
  
A problem on the distribution of a positive Borel measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r0825201.png" /> on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r0825202.png" /> of a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r0825203.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r0825204.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r0825205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r0825206.png" />, which generates a constant [[Newton potential|Newton potential]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r0825207.png" />, or constant [[Logarithmic potential|logarithmic potential]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r0825208.png" />, on any connected component of the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r0825209.png" />, i.e. the problem on the equilibrium distribution of an electric charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252010.png" /> on the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252011.png" /> of a conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252012.png" />.
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A problem on the distribution of a positive Borel measure $\lambda$ on the boundary $S$ of a compact set $K$ in the $n$-dimensional Euclidean space $\mathbf R^n$, $n\geq2$, which generates a constant [[Newton potential|Newton potential]] for $n\geq3$, or constant [[Logarithmic potential|logarithmic potential]] for $n=2$, on any connected component of the interior of $K$, i.e. the problem on the equilibrium distribution of an electric charge $\lambda(K)$ on the surface $S$ of a conductor $K$.
  
In the simplest classical case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252013.png" /> is a closed domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252014.png" /> homeomorphic to the sphere, bounded by a smooth simple surface or (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252015.png" />) by a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252016.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252019.png" />, the solution of Robin's problem is reduced to finding a non-trivial solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252021.png" />, of the homogeneous Fredholm-type integral equation of the second kind
+
In the simplest classical case when $K$ is a closed domain in $\mathbf R^n$ homeomorphic to the sphere, bounded by a smooth simple surface or (when $n=2$) by a curve $S$ of class $C^{1,\alpha}$, $0<\alpha<1$, $0\in K$, the solution of Robin's problem is reduced to finding a non-trivial solution $\nu(x)$, $x\in S$, of the homogeneous Fredholm-type integral equation of the second kind
  
<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/r/r082/r082520/r08252022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$\frac12\nu(x)+\frac{1}{k_n}\int\limits_S\nu(y)\frac{\partial}{\partial n_x}E_n(x,y)dS(y)=0,\quad x\in S,\tag{1}$$
  
 
under the normalization condition
 
under the normalization condition
  
<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/r/r082/r082520/r08252023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$\lambda(S)=\int\limits_S\nu(y)dS(y)=1.\tag{2}$$
  
 
Here
 
Here
  
<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/r/r082/r082520/r08252024.png" /></td> </tr></table>
+
$$E_2(x,y)=\ln\frac{1}{|x-y|},\quad E_n=\frac{1}{|x-y|^{n-2}}$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252026.png" /> is the distance between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252028.png" /> is the direction of the exterior normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252029.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252031.png" /> is the derivative, or density, of the absolutely-continuous measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252032.png" /> with respect to the Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252033.png" />,
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for $n\geq3$, $|x-y|$ is the distance between two points $x,y\in\mathbf R^n$, $n_x$ is the direction of the exterior normal to $S$ at the point $x\in S$, $\nu(x)$ is the derivative, or density, of the absolutely-continuous measure $\lambda$ with respect to the Lebesgue measure on $S$,
  
<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/r/r082/r082520/r08252034.png" /></td> </tr></table>
+
$$k_2=2\pi,\quad k_n=\frac{2(n-2)\pi^{n/2}}{\Gamma(n/2)}$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252035.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252036.png" /> is the area element of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252037.png" />. Equation (1) is obtained when one considers the interior [[Neumann problem(2)|Neumann problem]] for the domain bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252038.png" /> under vanishing boundary conditions, since the simple-layer potential
+
for $n\geq3$, and $dS(y)$ is the area element of the surface $S$. Equation \ref{1} is obtained when one considers the interior [[Neumann problem(2)|Neumann problem]] for the domain bounded by $S$ under vanishing boundary conditions, since the simple-layer potential
  
<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/r/r082/r082520/r08252039.png" /></td> </tr></table>
+
$$u(x)=u(x,K)=\int\limits_S\nu(y)E_n(x,y)dS(y),$$
  
called the Robin potential, equilibrium potential or [[Capacity potential|capacity potential]], should, according to the condition of Robin's problem, have a constant value on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252040.png" /> (see [[Potential theory|Potential theory]], and also [[#References|[2]]]). The solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252041.png" /> for the problem (1), (2) under the indicated conditions always exists in the class of continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252042.png" />. The measure
+
called the Robin potential, equilibrium potential or [[Capacity potential|capacity potential]], should, according to the condition of Robin's problem, have a constant value on $K$ (see [[Potential theory|Potential theory]], and also [[#References|[2]]]). The solution $\nu(x)$ for the problem \ref{1}, \ref{2} under the indicated conditions always exists in the class of continuous functions $C(S)$. The measure
  
<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/r/r082/r082520/r08252043.png" /></td> </tr></table>
+
$$\lambda(E)=\int\limits_E\nu(y)dS(y),\quad E\subset S,$$
  
which provides a solution of the Robin problem, is called the equilibrium measure. In a more complicated case, when the boundary of the compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252044.png" /> consists of a finite number of non-intersecting simple closed surfaces or (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252045.png" />) curves of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252047.png" /> (see [[#References|[2]]]), the Robin problem is solved in a similar way. Moreover, on bounded connected components of the open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252048.png" /> the Robin potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252049.png" /> also preserves its constant value, i.e. on the boundaries of these components the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252050.png" />.
+
which provides a solution of the Robin problem, is called the equilibrium measure. In a more complicated case, when the boundary of the compact set $K$ consists of a finite number of non-intersecting simple closed surfaces or (when $n=2$) curves of class $C^{1,\alpha}$, $0<\alpha<1$ (see [[#References|[2]]]), the Robin problem is solved in a similar way. Moreover, on bounded connected components of the open set $G=CK=\mathbf R^n\setminus K$ the Robin potential $u(x)$ also preserves its constant value, i.e. on the boundaries of these components the density $\nu(x)=0$.
  
Let the compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252051.png" /> be connected. The constant value of the Robin potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252052.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252053.png" />,
+
Let the compact set $K$ be connected. The constant value of the Robin potential $u(x)$ on $K$,
  
<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/r/r082/r082520/r08252054.png" /></td> </tr></table>
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$$\gamma=\int\limits_S\nu(y)E_n(x,y)dS(y),\quad x\in K,$$
  
is called the Robin constant of the compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252055.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252056.png" /> it is related to the harmonic, or Newton, [[Capacity|capacity]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252057.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252058.png" /> by the simple relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252059.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252061.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252062.png" />, the Robin constant can assume all values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252063.png" />; the harmonic capacity is then expressed by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252064.png" />.
+
is called the Robin constant of the compact set $K$. For $n\geq3$ it is related to the harmonic, or Newton, [[Capacity|capacity]] $C(K)$ of $K$ by the simple relation $C(K)=1/\gamma$; moreover, $0<\gamma<+\infty$, $0<C(K)<+\infty$. For $n=2$, the Robin constant can assume all values $-\infty<\gamma<+\infty$; the harmonic capacity is then expressed by the formula $C(K)=e^{-\gamma}$.
  
In another way, the equilibrium measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252065.png" /> is defined as the measure which yields the minimum of the energy integral
+
In another way, the equilibrium measure $\lambda$ is defined as the measure which yields the minimum of the energy integral
  
<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/r/r082/r082520/r08252066.png" /></td> </tr></table>
+
$$\iint\limits_{K\times K}E_n(x,y)d\mu(x)d\mu(y)$$
  
in the class of all measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252067.png" /> concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252068.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252070.png" />. Such a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252071.png" /> in the case of a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252072.png" /> with a smooth boundary coincides with the one found above, but it exists also in the general case of an arbitrary compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252074.png" />, if only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252075.png" />. The corresponding equilibrium potential
+
in the class of all measures $\mu$ concentrated on $K$ and such that $\mu\geq0$, $\mu(K)=1$. Such a measure $\lambda$ in the case of a compact set $K$ with a smooth boundary coincides with the one found above, but it exists also in the general case of an arbitrary compact set $K\subset\mathbf R^n$, $n\geq2$, if only $C(K)>0$. The corresponding equilibrium potential
  
<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/r/r082/r082520/r08252076.png" /></td> </tr></table>
+
$$u(x)=u(x;K)=\int E_n(x,y)d\lambda(y),$$
  
which is a generalization of the Robin potential, preserves the constant value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252077.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252078.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252079.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252080.png" />, everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082520/r08252081.png" /> except perhaps at the points of some set of capacity zero.
+
which is a generalization of the Robin potential, preserves the constant value $\gamma=1/C(K)$ for $n\geq3$, or $\gamma=-\ln C(K)$ for $n=2$, everywhere on $K$ except perhaps at the points of some set of capacity zero.
  
 
The name  "Robin problem"  is connected with studies of G. Robin (see [[#References|[1]]]).
 
The name  "Robin problem"  is connected with studies of G. Robin (see [[#References|[1]]]).

Revision as of 13:49, 20 August 2014

equilibrium problem, electrostatic problem

A problem on the distribution of a positive Borel measure $\lambda$ on the boundary $S$ of a compact set $K$ in the $n$-dimensional Euclidean space $\mathbf R^n$, $n\geq2$, which generates a constant Newton potential for $n\geq3$, or constant logarithmic potential for $n=2$, on any connected component of the interior of $K$, i.e. the problem on the equilibrium distribution of an electric charge $\lambda(K)$ on the surface $S$ of a conductor $K$.

In the simplest classical case when $K$ is a closed domain in $\mathbf R^n$ homeomorphic to the sphere, bounded by a smooth simple surface or (when $n=2$) by a curve $S$ of class $C^{1,\alpha}$, $0<\alpha<1$, $0\in K$, the solution of Robin's problem is reduced to finding a non-trivial solution $\nu(x)$, $x\in S$, of the homogeneous Fredholm-type integral equation of the second kind

$$\frac12\nu(x)+\frac{1}{k_n}\int\limits_S\nu(y)\frac{\partial}{\partial n_x}E_n(x,y)dS(y)=0,\quad x\in S,\tag{1}$$

under the normalization condition

$$\lambda(S)=\int\limits_S\nu(y)dS(y)=1.\tag{2}$$

Here

$$E_2(x,y)=\ln\frac{1}{|x-y|},\quad E_n=\frac{1}{|x-y|^{n-2}}$$

for $n\geq3$, $|x-y|$ is the distance between two points $x,y\in\mathbf R^n$, $n_x$ is the direction of the exterior normal to $S$ at the point $x\in S$, $\nu(x)$ is the derivative, or density, of the absolutely-continuous measure $\lambda$ with respect to the Lebesgue measure on $S$,

$$k_2=2\pi,\quad k_n=\frac{2(n-2)\pi^{n/2}}{\Gamma(n/2)}$$

for $n\geq3$, and $dS(y)$ is the area element of the surface $S$. Equation \ref{1} is obtained when one considers the interior Neumann problem for the domain bounded by $S$ under vanishing boundary conditions, since the simple-layer potential

$$u(x)=u(x,K)=\int\limits_S\nu(y)E_n(x,y)dS(y),$$

called the Robin potential, equilibrium potential or capacity potential, should, according to the condition of Robin's problem, have a constant value on $K$ (see Potential theory, and also [2]). The solution $\nu(x)$ for the problem \ref{1}, \ref{2} under the indicated conditions always exists in the class of continuous functions $C(S)$. The measure

$$\lambda(E)=\int\limits_E\nu(y)dS(y),\quad E\subset S,$$

which provides a solution of the Robin problem, is called the equilibrium measure. In a more complicated case, when the boundary of the compact set $K$ consists of a finite number of non-intersecting simple closed surfaces or (when $n=2$) curves of class $C^{1,\alpha}$, $0<\alpha<1$ (see [2]), the Robin problem is solved in a similar way. Moreover, on bounded connected components of the open set $G=CK=\mathbf R^n\setminus K$ the Robin potential $u(x)$ also preserves its constant value, i.e. on the boundaries of these components the density $\nu(x)=0$.

Let the compact set $K$ be connected. The constant value of the Robin potential $u(x)$ on $K$,

$$\gamma=\int\limits_S\nu(y)E_n(x,y)dS(y),\quad x\in K,$$

is called the Robin constant of the compact set $K$. For $n\geq3$ it is related to the harmonic, or Newton, capacity $C(K)$ of $K$ by the simple relation $C(K)=1/\gamma$; moreover, $0<\gamma<+\infty$, $0<C(K)<+\infty$. For $n=2$, the Robin constant can assume all values $-\infty<\gamma<+\infty$; the harmonic capacity is then expressed by the formula $C(K)=e^{-\gamma}$.

In another way, the equilibrium measure $\lambda$ is defined as the measure which yields the minimum of the energy integral

$$\iint\limits_{K\times K}E_n(x,y)d\mu(x)d\mu(y)$$

in the class of all measures $\mu$ concentrated on $K$ and such that $\mu\geq0$, $\mu(K)=1$. Such a measure $\lambda$ in the case of a compact set $K$ with a smooth boundary coincides with the one found above, but it exists also in the general case of an arbitrary compact set $K\subset\mathbf R^n$, $n\geq2$, if only $C(K)>0$. The corresponding equilibrium potential

$$u(x)=u(x;K)=\int E_n(x,y)d\lambda(y),$$

which is a generalization of the Robin potential, preserves the constant value $\gamma=1/C(K)$ for $n\geq3$, or $\gamma=-\ln C(K)$ for $n=2$, everywhere on $K$ except perhaps at the points of some set of capacity zero.

The name "Robin problem" is connected with studies of G. Robin (see [1]).

References

[1] G. Robin, "Sur la distribution de l'électricité à la surface des conducteurs fermés et des conducteurs ouverts" Ann. Sci. Ecole Norm. Sup. , 3 (1886) pp. 31–358
[2] N.M. Günter, "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar (1967) (Translated from Russian)
[3] N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)
[4] W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976)


Comments

In [1] Robin reconsiders and generalizes a problem formulated by S. Poisson (1811).

References

[a1] M. Tsuji, "Potential theory in modern function theory" , Chelsea, reprint (1975)
How to Cite This Entry:
Robin problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Robin_problem&oldid=33036
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article