Quasi-exceptional domains

TL;DR

This paper introduces quasi-exceptional domains, extending the concept of exceptional domains by allowing different boundary constants, and explores their classification, properties, and new examples, linking them to fluid dynamics and Abelian differentials.

Contribution

It defines quasi-exceptional domains, provides a partial classification using Abelian differentials, and constructs new periodic examples generalizing known hollow vortex configurations.

Findings

Partial classification of quasi-exceptional domains.

Introduction of a new two-parameter family of periodic domains.

Connection established between these domains and fluid dynamics problems.

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.