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On a question of T. Sheil-Small regarding valency of harmonic maps

Daoud Bshouty, Abdallah Lyzzaik (2012)

Annales UMCS, Mathematica

The aim of this work is to answer positively a more general question than the following which is due to T. Sheil-Small: Does the harmonic extension in the open unit disc of a mapping f from the unit circle into itself of the form f(eit) = eiϕ(t), 0 ≤ t ≤ 2π, where ϕ is a continuously non-decreasing function that satisfies ϕ(2π)−ϕ(0) = 2Nπ, assume every value finitely many times in the disc?

On the Riemann-Hilbert problem in multiply connected domains

Vladimir Ryazanov (2016)

Open Mathematics

We proved the existence of multivalent solutions with the infinite number of branches for the Riemann-Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The theorem is formulated in terms of harmonic measure and principal asymptotic values. It is also given the corresponding reinforced criterion for domains with rectifiable boundaries stated in terms of the natural parameter and nontangential...

Une propriété de la compactification de Martin d'un domaine euclidien

Alano Ancona (1979)

Annales de l'institut Fourier

Si B est une boule ouverte contenue dans le domaine euclidien Ω , tout filtre sur B , tendant non tangentiellement vers un point de Ω B , converge vers un point minimal dans le compactifié de Martin de Ω . On donne une application, et une variante dans le cas plan, et on termine par un contre-exemple apportant une solution négative à un problème de R.S. Martin. L’idée générale de l’article est d’établir des variantes des inégalités de Harnack pour déterminer la frontière de Martin du domaine.

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