Small deformations of a near cylindrical tube for the Canham-Helfrich Energy with applications to biological membranes

In this article we develop a quadratic energy which approximates the Canham-Helfrich energy for a tube-like surface with clamped boundary and area constraint. The energy is suited to the study of small deformations of biological membranes where the deformations are induced by point forces or point constraints due to the cytoskeleton or a phase dependent spontaneous curvature. Since the deformations we consider are small, we may assume that the surface of interest is a graph over a fixed, undeformed surface. A Lagrangian and the associated Euler-Lagrange equations for the graph are derived. Well-posedness of the Euler-Lagrange equations in suitable spaces is shown. Finally, we provide some illustrative numerical examples.