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 $\mathbf{F}$ defined on this filament, we consider the problem of determining the tension $\sigma$ 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 H\"older spaces. We find that the tension determination problem admits a unique solution if and only if the closed filament is {\em not} a circle. Furthermore, we show that the tension $\sigma$ gains one derivative with respect to the imposed line force density $\mathbf{F}$, and show that the tangential and normal components of $\mathbf{F}$ affect the regularity of $\sigma$ 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.