Research Problems in Function Theory — Problem 6.117

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.

For a proper plane domain $U$, a Borel set $E\subseteq\partial U$, and $z\in U$, $\omega(z,E,U)$ denotes harmonic measure: the probability that planar Brownian motion begun at $z$ first exits $U$ through $E$. For an arbitrary set, use the associated outer measure; for sets off the boundary, use their intersection with $\partial U$. For an open connected $G\subseteq\mathbb C$ containing zero, axial symmetry means $[z,\overline z]\subseteq G$ for every $z\in G$. For $t>0$, let $G_t^h$ and $G_t^v$ be the zero-containing components of $G\cap\{\operatorname{Im}z<t\}$ and $G\cap\{\operatorname{Re}z<t\}$. Define \[h_G(t)=\omega(0,\partial G_t^h\cap\{\operatorname{Im}z=t\},G_t^h),\qquad v_G(t)=\omega(0,\partial G_t^v\cap\{\operatorname{Re}z=t\},G_t^v).\]

Statement recorded in the supplied source.

For axially symmetric domains $D_1,D_2$ containing zero, does each of the following separately imply $D_1=D_2$: $h_{D_1}(t)=h_{D_2}(t)$ for every $t>0$; $v_{D_1}(t)=v_{D_2}(t)$ for every $t>0$; or both equalities together? If either individual implication holds, how few values of $t$ suffice: for example, is an infinite set of values whose corresponding horizontal or vertical lines meet both domains enough, or is a dense set required? For each implication that holds, does it remain valid for all simply connected domains containing zero, or for all domains containing zero without that restriction? [S2]

Short English statement

Can horizontal or vertical harmonic-measure profiles identify a domain, how many samples suffice, and can the symmetry assumption be weakened or removed?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306117 (AMR-022-6117). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Problem 6.117, complete problem and update. https://arxiv.org/abs/1809.07200.