Research Problems in Function Theory — Problem 3.2

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 subharmonic function is an upper semicontinuous function with values in $[-\infty,\infty)$, not identically $-\infty$ on a connected component, that satisfies the mean-value inequality on every closed ball in its domain. A real harmonic function is a twice continuously differentiable function $u$ satisfying $\Delta u=0$. A path to infinity in $\mathbb R^m$ is a continuous $\gamma:[0,\infty)\to\mathbb R^m$ with $|\gamma(t)|\to\infty$. The source update attributes an affirmative answer to Fuglede and explains that smooth or piecewise-linear paths can be chosen.

Statement recorded in the supplied source.

For every $m\geq3$ and every nonconstant harmonic function $u:\mathbb R^m\to\mathbb R$, is there a path $\gamma$ to infinity such that $u(\gamma(t))\to+\infty$? [S2]

Short English statement

Does every nonconstant harmonic function in dimension at least three tend to positive infinity along a continuous path?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303002 (AMR-022-3002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (2018), Problem 3.2 and Update 3.2. https://arxiv.org/abs/1809.07200.