An integral case of the axisymmetric shape equation of open vesicles with free edges

The equilibrium shapes of lipid vesicles are governed by the general shape equation which is derived from the minimization of the Helfrich free energy and can be reduced to the Willmore equation in a special case. The general shape equation is a high-order nonlinear partial differential equation and it is very difficult to find analytical solution even in axisymmetric case, which is reduced to a second-order ordinary differential equation. In the traditional axisymmetric shape equation, the turning radius is the variable. Here we study the shape equation by choosing the tangential angle as the variable. In this case, the Willmore equation is reduced to the Bernoulli differential equation and the general solution is obtained conveniently. We find that the curvature in this solution is discontinuous in some cases, which was not noticed previously. This solution can satisfy the boundary conditions for an open vesicle with free edges.