  The differential-geometric structure of the manifold of smooth shapes is
applied to the theory of shape optimization problems. In particular, a
Riemannian shape gradient with respect to the first Sobolev metric and the
Steklov-Poincar\'{e} metric are defined. Moreover, the covariant derivative
associated with the first Sobolev metric is deduced in this paper. The explicit
expression of the covariant derivative leads to a definition of the Riemannian
shape Hessian with respect to the first Sobolev metric. In this paper, we give
a brief overview of various optimization techniques based on the gradients and
the Hessian. Since the space of smooth shapes limits the application of the
optimization techniques, this paper extends the definition of smooth shapes to
$H^{1/2}$-shapes, which arise naturally in shape optimization problems. We
define a diffeological structure on the new space of $H^{1/2}$-shapes. This can
be seen as a first step towards the formulation of optimization techniques on
diffeological spaces.
