Asymptotic ratio of harmonic measures of sides of a boundary slit

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Let l be a Jordan arc in the upper half-plane ℍ with the initial point at z = 0. For ζ ∈ l, let ω+(a, ζ) and ω(a, ζ) denote the harmonic measures of the left and right shores of the slit lζ⊂ ℍ along the portion of the arc l travelled from 0 to ζ. In one of the seminars within the “Complex Analysis and Integrable Systems” semester held at the Mittag-Leffler Institute in 2011, D. Prokhorov suggested a study of the limit behavior of the quotient ω(a, ζ)/ω+(a, ζ) as ζ → 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 ω+(a, ζ)/ω(a, ζ) 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 Löwner equation are also discussed.

Original languageEnglish
Title of host publicationTrends in Mathematics
PublisherSpringer International Publishing
Number of pages27
StatePublished - 2018

Publication series

NameTrends in Mathematics
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X


  • Boundary slits
  • Harmonic measure
  • Löwner equation


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