Research Problems in Function Theory — Problem 9.10

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 $F\subset\mathbb R^n$ be closed, possibly unbounded. A real function on an open set is harmonic if it is twice continuously differentiable and its Laplacian is zero. Call $F$ a set of harmonic approximation if for every $g\in C(F;\mathbb R)$ harmonic on $F^{\circ}$ and every $\varepsilon>0$, there is a function $h$ harmonic on all of $\mathbb R^n$ with \[\sup_{x\in F}|g(x)-h(x)|<\varepsilon.\]

Statement recorded in the supplied source.

Give necessary and sufficient conditions on $F$ for it to be a set of harmonic approximation in every dimension $n\ge2$. [S2]

Short English statement

Characterize the closed sets on which continuous interior-harmonic functions admit arbitrarily accurate uniform approximation by globally harmonic functions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2309010 (AMR-022-9010). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 9.10, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.