Well-Posedness of the 3D Peskin Problem

This paper introduces the 3D Peskin problem: a two-dimensional elastic membrane immersed in a three-dimensional steady Stokes flow. We obtain the equations that model this free boundary problem and show that they admit a boundary integral reduction, providing an evolution equation for the elastic interface. We consider general nonlinear elastic laws, i.e., the fully nonlinear Peskin problem, and prove that the problem is well-posed in low-regularity H\"older spaces. Moreover, we prove that the elastic membrane becomes smooth instantly in time.