Research Problems in Function Theory — Problem 3.10

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.

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 $B=\{x\in\mathbb R^3:|x|<1\}$. The condition $u\in C^\infty(\overline B)$ means that derivatives of every order extend continuously to the closed ball. At $x\in\partial B$, the outward normal derivative is $\partial_\nu u(x)=\nabla u(x)\cdot x$. The source update reports that Wolff constructed counterexamples to this historical question.

Statement recorded in the supplied source.

If a real harmonic $u\in C^\infty(\overline B)$ has both $u=0$ and $\partial_\nu u=0$ on a subset $E\subset\partial B$ of positive surface area, must $u\equiv0$? [S2]

Short English statement

Do zero boundary values and zero normal derivatives on a positive-area part of the three-dimensional sphere force a smooth harmonic function to vanish?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303010 (AMR-022-3010). 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 3.10 and Update 3.10. https://arxiv.org/abs/1809.07200.