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 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
Pages273-299
Number of pages27
Edition9781493976584
DOIs
StatePublished - 2018

Publication series

NameTrends in Mathematics
Number9781493976584
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X

Keywords

  • Boundary slits
  • Harmonic measure
  • Löwner equation

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