Research Problems in Function Theory — Problem 7.48

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.

A domain $D\subset\mathbb R^n$ is linearly accessible if for every $y\notin D$ there is a unit vector $v$ for which the ray $\{y+sv:s\ge0\}$ avoids $D$. Suppose that $D$ is a linearly accessible Green domain, meaning it admits a positive Green function $g_D(x,x_0)$ for the Laplacian, zero on the boundary in the potential-theoretic sense, with pole at $x_0\in D$. Set $g_D(x_0,x_0)=+\infty$ and \[D_t=\{x\in D:g_D(x,x_0)>t\}.\]

Statement recorded in the supplied source.

For every dimension $n\ge3$, every such $D$, every pole $x_0\in D$ and every $t>0$, is $D_t$ linearly accessible? [S2] The source records that the corresponding preservation property holds in two dimensions. [S2]

Short English statement

Do all positive Green-function superlevel domains inherit linear accessibility in dimensions at least three?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307048 (AMR-022-7048). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.48 and its complete update. https://arxiv.org/abs/1809.07200.