We consider general weighted surface energies, where the energies have the form of weighted integrals over a closed surface and the weight depends on the normal and the mean curvature of the surface. Energies of this form have applications in many areas, such as material science, biology and image processing. Often one is interested in a minimum of such an energy. Then the first step in investigating this is the computation of the first variation of the energy with respect to perturbations of the surface. In this paper, we derive the first variation of the general weighted surface energy. We obtain this result using tools from shape differential calculus. Our result is valid for surfaces in any number of dimensions and unifies all previous results derived for specific examples such energies.
Citation: Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique Et Analyse Numerique
Pub Type: Journals
mean curvature, shape derivative, surface energy, Willmore functional