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arXiv · 1405.7754

Quasi-exceptional domains

Alexandre Eremenko·Erik Lundberg
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Abstract

Exceptional domains are domains on which there exists a positive harmonic function, zero on the boundary and such that the normal derivative on the boundary is constant. Recent results classify exceptional domains as belonging to either a certain one-parameter family of simply periodic domains or one of its scaling limits. We introduce quasi-exceptional domains by allowing the boundary values to be different constants on each boundary component. This relaxed definition retains the interesting property of being an \emph{arclength quadrature domain}, and also preserves the connection to the hollow vortex problem in fluid dynamics. We give a partial classification of such domains in terms of certain Abelian differentials. We also provide a new two-parameter family of periodic quasi-exceptional domains. These examples generalize the hollow vortex array found by Baker, Saffman, and Sheffield (1976). A degeneration of regions of this family provide doubly-connected examples.

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Alexandre Eremenko, Erik Lundberg. 2014-05-30. Quasi-exceptional domains. https://doi.org/10.2140/pjm.2015.276.167

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