Research Problems in Function Theory — Problem 3.22

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.

Write $\mathbb D=\{z\in\mathbb C:|z|<1\}$ and $\mathbb T=\partial\mathbb D$. Let $D\subset\mathbb D$ be a connected open domain containing $0$, with boundary consisting of the outer circle $\mathbb T$ and an inner set $E\subset\mathbb D$ that is relatively closed in $\mathbb D$. Require that every radius $\{te^{i\theta}:0<t<1\}$ meets $E$. Let $\omega_D(0,\mathbb T)$ be harmonic measure of the outer boundary, equivalently the probability that Brownian motion starting at $0$ exits $D$ through $\mathbb T$ rather than through $E$. Define \[c_* = \sup_{D,E}\omega_D(0,\mathbb T),\] where the supremum is over all pairs with these properties.

Statement recorded in the supplied source.

Determine the exact value of $c_*$. The inner boundary $E$ is not required to be connected; the special case of an inner continuum is only one part of the problem. [S2]

Short English statement

Find the largest possible chance of escaping through the unit circle when an inner closed boundary meets every radius, allowing arbitrary closed inner sets.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303022 (AMR-022-3022). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory, arXiv:1809.07200, 2018. Problem 3.22, its complete Update 3.22, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.