Research Problems in Function Theory — Problem 3.12

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 $\Delta u=0$. Let $m\geq3$, let $P=\{x\in\mathbb R^m:x_m=0\}$, and let $E\subset P$ be closed such that $D=\mathbb R^m\setminus E$ is a connected domain of infinite connectivity (its complement has infinitely many connected components). Suppose there are fixed $R,\varepsilon>0$ such that every $(m-1)$-dimensional disk $B_P(a,R)$ in $P$ satisfies \[\mathcal L^{m-1}(E\cap B_P(a,R))\geq\varepsilon.\] Let $u$ be positive and harmonic on $D$, extend continuously to all of $\mathbb R^m$, and vanish on $E$.

Statement recorded in the supplied source.

For $m=3$, must there exist a constant $c$ and a function $\phi$ uniformly bounded on $\{x_m>0\}$ such that \[u(x)=c x_m+\phi(x)\qquad(x_m>0)?\] Does the same assertion hold for every $m>3$ with $(m-1)$-dimensional measure in the density hypothesis? [S2]

Short English statement

Does a uniformly dense planar zero set force a positive harmonic function to equal a linear function plus a bounded remainder in the upper half-space, in three and higher dimensions?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303012 (AMR-022-3012). 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 3.12. https://arxiv.org/abs/1809.07200.