The Euler–Helfrich functional

Abstract

We investigate equilibrium configurations for surface energies which contain the squared \(L^2\) norm of the difference of the mean curvature H and the spontaneous curvature \(c_o\) coupled with the elastic energy of the boundary curve, which we studied previously in Palmer and Pámpano (J Nonlinear Sci 31(1):23, 2021). It is shown that if a critical surface for this type of functional is axially symmetric, then it satisfies a simpler second order variational problem. Many examples of solutions of this are given.