Research Problems in Function Theory — Problem 7.26

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.

Let $B^3=\{x=(x_1,x_2,x_3)\in\mathbb R^3:|x|<1\}$. A real function $u$ is harmonic on $B^3$ if it is twice continuously differentiable and $\Delta u=\sum_{j=1}^3\partial^2u/\partial x_j^2=0$. A homeomorphism is a continuous bijection with continuous inverse.

Statement recorded in the supplied source.

Do there exist harmonic functions $u_1,u_2,u_3:B^3\to\mathbb R$ such that \[x\longmapsto (u_1(x),u_2(x),u_3(x))\] is a homeomorphism from $B^3$ onto all of $\mathbb R^3$? [S2] The source distinguishes the three-dimensional question from Radó's negative result in two dimensions. [S2]

Short English statement

Can the three coordinates of a homeomorphism from the open three-dimensional ball onto Euclidean space all be harmonic?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307026 (AMR-022-7026). CC BY4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter7, Problem7.26 and its complete update. https://arxiv.org/abs/1809.07200.