The tension determination problem for an inextensible interface in 2D Stokes flow

Consider an inextensible closed filament immersed in a 2D Stokes fluid. Given a force density F defined on this filament, we consider the problem of determining the tension σ on this filament that ensures the filament is inextensible. This is a subproblem of dynamic inextensible vesicle and membrane problems, which appear in engineering and biological applications. We study the well-posedness and regularity properties of this problem in Hölder spaces. We find that the tension determination problem admits a unique solution if and only if the closed filament is not a circle. Furthermore, we show that the tension σ gains one derivative with respect to the imposed line force density F and show that the tangential and normal components of F affect the regularity of σ in different ways. We also study the near singularity of the tension determination problem as the interface approaches a circle and verify our analytical results against numerical experiment.