We intend to lay the mathematical foundations for a rigorous non-smooth shape calculus. A typical application area are geometric inverse problems, which often involve partial differential equations. Surface fairing with edge preservation as well as the detection of non-smooth inclusions through remote sensing and tomography are typical examples of problems greatly benefiting from this research.

We shall introduce new geometric functionals, which allow a fine-grained control over the non-smooth features of desired shapes. As a particular example, we mention the total surface variation of the normal vector field. To this end, we extend the concept of the total variation semi-norm to nonsmooth functions and geometric quantities on non-smooth surfaces. This novel approach will also allow for an anisotropic control of preferred shapes.

Hand in hand with the theoretical considerations above, we will also focus on consistent discrete realizations. In view of the fact that any triangulated surface is essentially non-smooth, we expect considerable improvements of the present state of the art in computational shape optimization. For example, we will address the question of finding the best possible curvature approximation consistent with the tangential Stokes formula.

To exemplify the benefits of our novel approach, we intend to solve a number of prototypical application problems of increasing complexity, in particular problems in surface fairing, inverse obstacle problems, electrical impedance tomography and inverse electro-magnetic scattering problems governed by Maxwell's equations.