On the Symmetry of Second Derivatives in Optimal Shape Design and Suucient Optimality Conditions for Shape Functionals

For some heuristic approaches of the boundary variation in shape optimization the computation of second derivatives of domain and boundary integral functionals, their symmetry and a comparison to the velocity eld or material derivative method are discussed. Moreover, for some of these approaches the funcionals are Fr echet-diierentiable, because an embedding into a Banach space problem is possible. This allows the discussion of suucient condition in terms of a coercivity assumption on the second Fr echet-derivative. The theory is illustrated by a discussion of the famous Dido problem.