# 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)