The shape derivative of the Gauss curvature

We present a review of results about the shape derivatives of scalar- and vector-valued shape functions, and extend the results from Dogan and Nochetto [ESAIM Math. Model. Numer. Anal. 46 (2012), no. 1, 59-79] to more general surface energies. In that article, Dogan and Nochetto consider surface energies defined as integrals over surfaces of functions that can depend on the position, the unit normal and the mean curvature of the surface. In this work we present a systematic way to derive formulas for the shape derivative of more general geometric quantities, including the Gauss curvature (a new result not available in the literature) and other geometric invariants (eigenvalues of the second fundamental form). This is done for hyper-surfaces in the Euclidean space of any finite dimension. As an application of the results, with relevance for numerical methods in applied problems, we derive a Newton-type method to approximate a minimizer of a shape functional. We finally find the particular formulas for the first and second order shape derivatives of the area and the Willmore functional, which are necessary for the aforementioned Newton-type method.