Robin problem

equilibrium problem, electrostatic problem

A problem on the distribution of a positive Borel measure on the boundary of a compact set in the -dimensional Euclidean space , , which generates a constant Newton potential for , or constant logarithmic potential for , on any connected component of the interior of , i.e. the problem on the equilibrium distribution of an electric charge on the surface of a conductor .

In the simplest classical case when is a closed domain in homeomorphic to the sphere, bounded by a smooth simple surface or (when ) by a curve of class , , , the solution of Robin's problem is reduced to finding a non-trivial solution , , of the homogeneous Fredholm-type integral equation of the second kind

(1)

under the normalization condition

(2)

Here

for , is the distance between two points , is the direction of the exterior normal to at the point , is the derivative, or density, of the absolutely-continuous measure with respect to the Lebesgue measure on ,

for , and is the area element of the surface . Equation (1) is obtained when one considers the interior Neumann problem for the domain bounded by under vanishing boundary conditions, since the simple-layer potential

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

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 consists of a finite number of non-intersecting simple closed surfaces or (when ) curves of class , (see [2]), the Robin problem is solved in a similar way. Moreover, on bounded connected components of the open set the Robin potential also preserves its constant value, i.e. on the boundaries of these components the density .

Let the compact set be connected. The constant value of the Robin potential on ,

is called the Robin constant of the compact set . For it is related to the harmonic, or Newton, capacity of by the simple relation ; moreover, , . For , the Robin constant can assume all values ; the harmonic capacity is then expressed by the formula .

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

in the class of all measures concentrated on and such that , . Such a measure in the case of a compact set with a smooth boundary coincides with the one found above, but it exists also in the general case of an arbitrary compact set , , if only . The corresponding equilibrium potential

which is a generalization of the Robin potential, preserves the constant value for , or for , everywhere on 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)