Harmonic measure of arcs of fixed length

HendaHewa Supem Ranjana Samarasiri, Alexander Solynin

Research output: Contribution to journalArticlepeer-review

Abstract

Abstract. In this paper, we consider Jordan domains $\Omega$ with piece-wise smooth boundaries such that all arcs $\alpha\subset \partial \Omega$ having fixed length $l$, $0<l<{\mbox{length}}(\partial \Omega)$, have equal harmonic measures $\omega(z_0,\alpha,\Omega)$ evaluated at some point $z_0\in \Omega$. We prove that $\Omega$ is a disk centered at $z_0$ if the ratio $l/{\mbox{length}}(\partial \Omega)$ is irrational and that $\Omega$ possesses rotational symmetry by some angle $2\pi/n$, $n\ge 2$, around the point $z_0$, if this ratio is rational.
Original language English 145-152 Zapiski POMI Published - Oct 15 2020