The classification problem for arclength null quadrature domains

A planar domain is referred to as an arclength null quadrature domain if the integral along the boundary of any analytic in the domain function, representable by the Cauchy integral of its boundary values ( Smirnov class $E^1$), vanishes. Obviously, all such domains must be unbounded. We prove the existence of a "roof function" (a positive harmonic function whose gradient coincides with the inward pointing unit normal along the boundary) for arclength null quadrature domains having finitely many boundary components. This bridges a gap toward classification of arclength null quadrature domains by removing an a priori assumption from previous classification results, in particular, it completes the theorem of Khavinson-Lundberg Teodorescu from 2012 that established that domains allowing "roof functions" are arclength null quadrature domains . This result also strengthens an existing connection to free boundary problems for Laplace's equation and the hollow vortex problem in fluid dynamics. The proof is based on the techniques originated in classical works of Ahlfors, Carleman and Denjoy. We shall also discuss the current status of the classification problem for arclength null quadrature domains. This is a 2021 joint work with Erik Lundberg.
This is a 2021 joint work with Erik Lundberg allowing to reduce the problem of classification of arclength quadrature domains to that of domains admitting a positive harmonic function with locally constant boundary values with a constant normal derivative on the boundary, so-called quasi-exceptional domains.