First variation of the general curvature-dependent surface energy

We consider general surface energies, which are weighted integrals over a closed surface with a weight function depending on the position, the unit normal and the mean curvature of the surface. Energies of this form have applications in many areas, such as materials science, biology and image processing. Often one is interested in finding a surface that minimizes such an energy, which entails finding its first variation with respect to perturbations of the surface. We present a concise derivation of the first variation of the general surface energy using tools from shape differential calculus. We first derive a scalar strong form and next a vector weak form of the first variation. The latter reveals the variational structure of the first variation, avoids dealing explicitly with the tangential gradient of the unit normal, and thus can be easily discretized using parametric finite elements. Our results are valid for surfaces in any number of dimensions and unify all previous results derived for specific examples of such surface energies.