Second derivative and sufficient optimality conditions for shape functionals

For some heuristic approaches to boundary variation in shape optimization the computation of second derivatives of domain and boundary integral functionals, their symmetry and a comparison with the velocity field or material derivative method are discussed. Moreover, for these approaches the functionals are Fréchet differentiable in some sense, because at least a local embedding into a Banach space problem is possible. This allows the discussion of sufficient conditions in terms of a coercivity assumption on the second Fréchet derivative. The theory is illustrated by the discussion of the classical Dido problem.