Research Problems in Function Theory — Problem 3.17

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 $D'\subset D$ be Lipschitz domains in $\mathbb R^n$, $n\geq3$: locally their boundaries are graphs of Lipschitz functions. Suppose $\partial D'\cap\partial D$ is a compact subset of the relative interior of $\Gamma\subset\partial D$. Fix $P_0\in D'$. Let $\mathcal H$ consist of positive harmonic functions on $D$ that extend continuously with value $0$ on $\Gamma$ and satisfy $h(P_0)=1$. The source update attributes affirmative boundary-Harnack results to Dahlberg and Wu.

Statement recorded in the supplied source.

Is there a finite constant $C$, depending only on $D,D',\Gamma,P_0$ and $n$, such that \[h_1(P)\leq C h_2(P)\qquad(P\in D',\ h_1,h_2\in\mathcal H)?\] [S2]

Short English statement

Are all normalized positive harmonic functions that vanish on the same boundary portion uniformly comparable throughout the smaller Lipschitz domain?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303017 (AMR-022-3017). 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 3.17 and Update 3.17. https://arxiv.org/abs/1809.07200.