Robin constant

Remarks

The logarithmic capacity of $K$ is $\operatorname{cap}(K) = e^{-V(K)}$. If $V(K) < \infty$ and $\Omega$ is the unbounded component of $\hat\C \setminus K$, the Green function of $\Omega$ with pole at infinity satisfies $g_\Omega(z, \infty) = \log \abs{z} + V(K) + o(1)$ as $z \to \infty$.

Example: let $K$ be the circle of radius $r$ centred at $0$ and let $\mu$ be normalized arc length on it. For $\abs{z} = r$ the mean value of $\log(1/\abs{z - w})$ over the circle is $-\log r$, so $I(\mu) = -\log r$. This $\mu$ is the minimizer, so $V(K) = -\log r$. The Green function $\log(\abs{z}/r)$ of the outside of the circle agrees with the expansion above.

Non-example of a finite value: for $K = \set{0}$ the only probability measure is $\delta_0$, and $I(\delta_0) = \log(1/0) = \infty$, so $V(K) = \infty$.

Some authors use the opposite sign or the energy kernel $\log \abs{z - w}$, so that the Robin constant becomes $\log \operatorname{cap}(K)$.