To deal with sufficient conditions of optimality in shape optimization, some difficulties arise. One the one hand, one cannot avoid a Fréchet-calculus of second order. To obtain a related Banach space embedding at least for a subclass of domains, we will investigate a special boundary variational approach together with a potential ansatz for the solution of the state equation. On the other hand, different kinds of the two-norm discrepancy occur for studying sufficient conditions for boundary and domain shape functionals as well as for more interesting shape optimization problems. To ensure optimality, this requires a more detailed treatment of related second order remainders. The Dido problem and the torsional rigidity of an elastic bar will be discussed as examples. The last part of the talk will be devoted to some extensions with respect to the state equation and the class of domains under consideration.