Asymptotic ratio of harmonic measures of sides of a boundary slit

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Abstract

Let $l$ be a Jordan arc in the upper half-plane $\mathbb{H}$ with the initial point at $z=0$. For $\zeta\in l$, let $\omega^+(a,\zeta)$ and $\omega^-(a,\zeta)$ denote the harmonic measures of the left and right shores of the slit $l_\zeta\subset \mathbb{H}$ along the portion of the arc $l$ travelled from $0$ to $\zeta$. In one of the seminars within the ``\emph{Complex Analysis and Integrable Systems}'' semester held at the Mittag-Leffler Institut in 2011, D. Prokhorov suggested a study of the limit behavior of the quotient $\omega^-(a,\zeta)/\omega^+(a,\zeta)$ as $\zeta\to 0$ along $l$. In recent publications, D.~Prokhorov and his coauthors discussed this problem and proved several results for smooth slits. In this paper, we study the limit behavior of $\omega^+(a,\zeta)/\omega^-(a,\zeta)$ for a broader class of continuous slits which includes radially and angularly oscillating slits. Some related questions concerning behavior of the driving term of the corresponding chordal L\"{o}
Original languageEnglish
Title of host publicationAsymptotic ratio of harmonic measures of sides of a boundary slit
PublisherBirkhäuser, Springer International Publishing AG
Pages21pp.
StatePublished - Apr 2018

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