Research Problems in Function Theory — Problem 3.28

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 $n\ge2$ and let $\Omega\subset\mathbb R^n$ be a nonempty connected open set, without any finite-volume or boundedness assumption. Write \[\mathcal H^1(\Omega)=\{h\in C^2(\Omega):\Delta h=0,\ \int_\Omega|h(x)|\,dx<\infty\}. \] Lebesgue integration is used.

Statement recorded in the supplied source.

Classify all domains $\Omega$ such that \[\int_\Omega h(x)\,dx=0\qquad\text{for every }h\in\mathcal H^1(\Omega). \] Include every dimension $n\ge2$; the source records an existing planar classification. [S2]

Short English statement

Classify all domains on which every integrable harmonic function has integral zero, including the higher-dimensional cases.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303028 (AMR-022-3028). 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 3.28, its complete Update 3.28, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.