Research Problems in Function Theory — Problem 3.15

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 $D\subset\mathbb C$ be a doubly connected domain with boundary curves $\alpha$ and $\beta$, let $z_0,z_1\in D$, and let $A,B\in\mathbb R$. Boundary values mean the ordinary limiting values from $D$. Let $\mathcal U$ consist of subharmonic functions $u$ on $D$ with boundary values $A$ on $\alpha$, $B$ on $\beta$, for which there is a continuous curve $\gamma:[0,1]\to D\cup\alpha$ with $\gamma(0)=z_1$, $\gamma(1)\in\alpha$, $\gamma([0,1))\subset D$, and $u\leq0$ on $\gamma([0,1))$.

Statement recorded in the supplied source.

Determine $\sup_{u\in\mathcal U}u(z_0)$, whether it is attained, and the maximizing functions, for the given $D,\alpha,\beta,z_0,z_1,A,B$; the curve may depend on $u$. If $\mathcal U$ is empty, the supremum is $-\infty$. [S2]

Short English statement

Maximize a subharmonic value in a doubly connected domain with fixed constants on the two boundaries and a nonpositive path from a second point to the first boundary.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303015 (AMR-022-3015). 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 3.15. https://arxiv.org/abs/1809.07200.