Research Problems in Function Theory — Problem 7.71

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.

Fix $n\ge2$. A domain of holomorphy $\Omega\subset\mathbb C^n$ is a connected open set that is a domain of existence for a holomorphic function. A holomorphic function $p$ has no continuation through a boundary point $a$ if there is no neighborhood of $a$ carrying a holomorphic function agreeing with $p$ on an adjoining open part of $\Omega$. The complex Laplace operator is \[\Delta_{\mathbb C}p=\sum_{j=1}^n\frac{\partial^2p}{\partial z_j^2}, \] which is different from the real Laplacian or the sum of mixed $z_j,\bar z_j$ derivatives.

Statement recorded in the supplied source.

Find necessary and sufficient conditions on $\Omega$ for the existence of $p\in\mathcal O(\Omega)$ satisfying $\Delta_{\mathbb C}p=0$ and admitting no analytic continuation through its boundary. The question is for arbitrary domains of holomorphy in every dimension $n\ge2$. [S2]

Short English statement

Which domains of holomorphy admit a solution of the complex Laplace equation whose analytic continuation stops at the boundary?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307071 (AMR-022-7071). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.71, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.