Research Problems in Function Theory — Problem 3.24

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 real-valued function $u\in C^2(\mathbb R^3)$ is harmonic on all of $\mathbb R^3$ if \[\Delta u=\frac{\partial^2u}{\partial x_1^2}+\frac{\partial^2u}{\partial x_2^2}+\frac{\partial^2u}{\partial x_3^2}=0. \] For $p>0$, let $C_p=\{x\in\mathbb R^3:x_1^2+x_2^2=px_3^2\}$, including both sheets and the vertex.

Statement recorded in the supplied source.

Determine all $p>0$ for which there is a nonzero harmonic function $u$ on the whole of $\mathbb R^3$ such that $u(x)=0$ for every $x\in C_p$. [S2]

Short English statement

Which circular cones are contained in the zero set of a nontrivial harmonic function defined everywhere in three-dimensional space?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303024 (AMR-022-3024). 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 3.24, its complete Update 3.24, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.