Riemann–Hilbert theory for problems with vanishing coefficients that arise in nonlinear hydrodynamics

Motivated by a question from mathematical hydrodynamics, this paper studies the solution set of Riemann–Hilbert problems on the unit disc D in C of the formψ=aϕ̄ on ∂D,where ϕ,ψ belong to subclasses of N+, the Nevanlinna–Smirnov functions on D, and the coefficient a is a real-valued non-negative function which vanishes at points of ∂D.