Research Problems in Function Theory — Problem 7.45

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Fix $p\in\mathbb N$ and a slit length $\ell\in(0,1)$. For distinct angles $\theta_1,\ldots,\theta_p$ modulo $2\pi$, set \[D_\theta=\mathbb D\setminus\bigcup_{j=1}^p\{te^{i\theta_j}:1-\ell\le t<1\}.\] Let $u_\theta(z)$ be the harmonic measure at $z\in D_\theta$ of the outer circle $\partial\mathbb D$; the two sides of each slit are assigned boundary value zero. In particular $u_\theta(0)$ is the probability that planar Brownian motion starting at zero exits through the outer circle rather than a slit.

Statement recorded in the supplied source.

For each fixed $p$ and $\ell$, determine the configurations of the $p$ radial slits that minimise $u_\theta(0)$. [S2] Update7.45 reports Dubinin's proof that equally spaced slits minimise the quantity. [S2]

Short English statement

Arrange a fixed number of equal radial slits to minimise the harmonic measure of the outer boundary seen from the centre.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307045 (AMR-022-7045). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.45 and its complete update. https://arxiv.org/abs/1809.07200.