Robin constant

A numerical characteristic of a set of points in a Euclidean space , , closely connected with the capacity of the set.

Let be a compact set in , and let be a positive Borel measure concentrated on and normalized by the condition . The integral

where

and is the distance between two points , is the energy of (cf. Energy of measures). The Robin constant of the compact set is the lower bound over all measures of the indicate type. If , then this bound is finite and is attained for some (unique) equilibrium, or capacitary, measure , , , concentrated on ; if , then for all measures of the indicated type. The Robin constant of is related to its capacity by the formula

If the boundary of is sufficiently smooth, for example, if it consists of a finite number of pairwise non-intersecting simple closed surfaces (for ) or curves (for ) of class , , then the equilibrium measure is concentrated on the part which forms the boundary of that connected component of the complement which contains the point at infinity. The equilibrium potential, Robin potential or capacity potential, i.e. the potential of the equilibrium measure

in this case assumes a constant value on , equal to , which allows one to calculate the Robin constant of a compact set in the simplest cases (see Robin problem). For instance, the Robin constant of a disc of radius in is , and the Robin constant of a ball of radius in , , is . In the case of an arbitrary compact set of positive capacity, everywhere and everywhere on the support of the equilibrium measure , except possibly at the points of some polar set; moreover, .

Let be a domain in the extended complex plane containing inside it the point at infinity and having a Green function with pole at infinity. Then the following representation holds:

(1)

where is a complex variable, is the Robin constant of the domain and is a harmonic function in ; moreover,

The Robin constant of the domain , defined by (1), coincides with the Robin constant of the compact set : . If the Green function for the domain does not exist, then one assumes that .

By generalizing the representation (1) to a Riemann surface which has a Green function, one can obtain a local representation of the Green function with pole :

(2)

where is a local uniformizing parameter in a neighbourhood of the pole , , is the Robin constant of the Riemann surface relative to the pole , and is a harmonic function in a neighbourhood of ; moreover, . For Riemann surfaces which do not have a Green function one assumes . In expression (2) the value of the Robin constant depends now on the choice of the pole . However, the relations and are independent of the choice of the pole. This allows one to use the notion of a Robin constant in the classification of Riemann surfaces (cf. Riemann surfaces, classification of).

References

[1]	R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German)
[2]	S. [S. Stoilov] Stoilow, "Leçons sur les principes topologiques de la théorie des fonctions analytiques" , Gauthier-Villars (1938)
[3]	L. Sario, M. Nakai, "Classification theory of Riemann surfaces" , Springer (1970)

Comments

See also the references quoted in Capacity; Energy of measures; Robin problem.