View source for Robin 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" />.

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

<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>

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>

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>

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" />,

<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>

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

<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>

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

<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>

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" />.

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" />,

<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>

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" />.

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

<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>

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

<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>

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.

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.M. Günter,   "Potential theory and its applications to basic problems of mathematical physics" , F. Ungar  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.S. Landkof,   "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W.K. Hayman,   P.B. Kennedy,   "Subharmonic functions" , '''1''' , Acad. Press  (1976)</TD></TR></table>



====Comments====
In [[#References|[1]]] Robin reconsiders and generalizes a problem formulated by S. Poisson (1811).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Tsuji,   "Potential theory in modern function theory" , Chelsea, reprint  (1975)</TD></TR></table>