Research Problems in Function Theory — Problem 3.27

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 $n\ge2$ and an unbounded domain $D\subset\mathbb R^n$, where a domain is nonempty, open and connected. A real-valued harmonic function $u$ on $D$ satisfies $u\in C^2(D)$ and $\Delta u=0$. The Euclidean norm is denoted by $|x|$.

Statement recorded in the supplied source.

Does there exist, for every such $D$, a continuous function $\varepsilon_D:[0,\infty)\to(0,\infty)$ such that \[\left(|u(x)|<\varepsilon_D(|x|)\text{ for every }x\in D\right)\ \Longrightarrow\ u\equiv0 \] for every harmonic $u$ on $D$? The function may depend on $D$ and $n$, but not on $u$. [S2]

Short English statement

For each unbounded domain, can one choose a positive radial size bound that no nonzero harmonic function can satisfy everywhere?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303027 (AMR-022-3027). 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 3.27, its complete Update 3.27, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.