Robin constant

A numerical characteristic of a set of points in a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, closely connected with the capacity of the set.

Let $ K $ be a compact set in $ \mathbf R ^ {n} $, and let $ \mu $ be a positive Borel measure concentrated on $ K $ and normalized by the condition $ \mu ( K) = 1 $. The integral

$$ V ( \mu ) = {\int\limits \int\limits } _ {K \times K } E _ {n} ( x , y ) d \mu ( x) d \mu ( y) , $$

where

$$ E _ {2} ( x , y ) = \mathop{\rm ln} \frac{1}{| x - y | } ,\ \ E _ {n} ( x , y ) = \frac{1}{| x - y | ^ {n-2} } \textrm{ for } \ n \geq 3 , $$

and $ | x - y | $ is the distance between two points $ x , y \in \mathbf R ^ {n} $, is the energy of $ \mu $( cf. Energy of measures). The Robin constant of the compact set $ K $ is the lower bound $ \gamma ( K) = \inf V ( \mu ) $ over all measures $ \mu $ of the indicate type. If $ \gamma ( K) < + \infty $, then this bound is finite and is attained for some (unique) equilibrium, or capacitary, measure $ \lambda > 0 $, $ \gamma ( K) = V ( \lambda ) $, $ \lambda ( K) = 1 $, concentrated on $ K $; if $ \gamma ( K) = + \infty $, then $ V ( \mu ) = + \infty $ for all measures $ \mu $ of the indicated type. The Robin constant of $ K $ is related to its capacity by the formula

$$ \gamma ( K) = \frac{1}{C ( K) } \ \textrm{ for } n \geq 3 , $$

$$ \gamma ( K) = - \mathop{\rm ln} C ( K) \ \textrm{ for } n = 2 . $$

If the boundary $ S $ of $ K $ is sufficiently smooth, for example, if it consists of a finite number of pairwise non-intersecting simple closed surfaces (for $ n \geq 3 $) or curves (for $ n = 2 $) of class $ C ^ {1 , \alpha } $, $ 0 < \alpha < 1 $, then the equilibrium measure $ \lambda $ is concentrated on the part $ \widetilde{S} \subset S $ which forms the boundary of that connected component of the complement $ C K = \mathbf R ^ {n} \setminus K $ which contains the point at infinity. The equilibrium potential, Robin potential or capacity potential, i.e. the potential of the equilibrium measure

$$ u ( x) = \int\limits E _ {n} ( x , y ) d \lambda ( y) , $$

in this case assumes a constant value on $ \widetilde{S} $, equal to $ \gamma ( K) $, 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 $ r > 0 $ in $ \mathbf R ^ {2} $ is $ - \mathop{\rm ln} r $, and the Robin constant of a ball of radius $ r > 0 $ in $ \mathbf R ^ {n} $, $ n \geq 3 $, is $ 1 / r ^ {n-2} $. In the case of an arbitrary compact set $ K $ of positive capacity, $ u ( x) \leq \gamma ( K) $ everywhere and $ u ( x) = \gamma ( K) $ everywhere on the support $ S ( \lambda ) $ of the equilibrium measure $ \lambda $, except possibly at the points of some polar set; moreover, $ S ( \lambda ) \subset K $.

Let $ D $ be a domain in the extended complex plane $ \overline{\mathbf C}\; $ containing inside it the point at infinity and having a Green function $ g ( z , \infty ) $ with pole at infinity. Then the following representation holds:

$$ \tag{1 } g ( z , \infty ) = \mathop{\rm ln} | z | + \gamma ( D) + \epsilon ( z , \infty ) , $$

where $ z = x + i y $ is a complex variable, $ \gamma ( D) $ is the Robin constant of the domain $ D $ and $ \epsilon ( z , \infty ) $ is a harmonic function in $ D $; moreover,

$$ \lim\limits _ {| z | \rightarrow \infty } \epsilon ( z , \infty ) = 0 . $$

The Robin constant of the domain $ D $, defined by (1), coincides with the Robin constant of the compact set $ \partial D $: $ \gamma ( D) = \gamma ( \partial D ) $. If the Green function for the domain $ D $ does not exist, then one assumes that $ \gamma ( D) = + \infty $.

By generalizing the representation (1) to a Riemann surface $ R $ which has a Green function, one can obtain a local representation of the Green function $ g ( p , p _ {0} ) $ with pole $ p _ {0} \in R $:

$$ \tag{2 } g ( p , p _ {0} ) = \ \mathop{\rm ln} \frac{1}{| z - z _ {0} | } + \gamma ( R ; p _ {0} ) + \epsilon ( p , p _ {0} ) , $$

where $ z = z ( p) $ is a local uniformizing parameter in a neighbourhood of the pole $ p _ {0} $, $ z ( p _ {0} ) = z _ {0} $, $ \gamma ( R ; p _ {0} ) $ is the Robin constant of the Riemann surface $ R $ relative to the pole $ p _ {0} $, and $ \epsilon ( p , p _ {0} ) $ is a harmonic function in a neighbourhood of $ p _ {0} $; moreover, $ \lim\limits _ {p \rightarrow p _ {0} } \epsilon ( p , p _ {0} ) = 0 $. For Riemann surfaces $ R $ which do not have a Green function one assumes $ \gamma ( R ; p _ {0} ) = + \infty $. In expression (2) the value of the Robin constant $ \gamma ( R ; p _ {0} ) $ depends now on the choice of the pole $ p _ {0} \in R $. However, the relations $ \gamma ( R ; p _ {0} ) < + \infty $ and $ \gamma ( R ; p _ {0} ) = + \infty $ 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.