  In this thesis we develop a functional analytic framework for shape
optimization with elliptic partial differential equation (PDE) constraints in
classical function spaces (H\"older spaces). This approach is motivated by
shape optimization problems, which are subjected to linear elasticity
constraints and involve a special class of shape functionals which calculate
the failure rate of a mechanically loaded device w.r.t. the component shape.
These objectives are ill-defined for $H^1$-solutions of the state equation and
the shape derivatives are not defined for $H^1$-material derivatives. Thus, the
resulting optimal reliability problems can not be solved by the already
existing methods of shape calculus. We develop a general concept on Banach and
Hilbert spaces which is based on parameter depending variational equations,
classical PDE Solutions, Schauder estimates and compact embeddings and which
allows to transfer differentiability in lower Banach space topologies to higher
ones. We apply this framework to the linear elasticity equation and its
variational formulation, given that the domain is transformed according to the
speed method. We prove the existence of material and local shape derivatives in
H\"older spaces, the existence of shape derivatives and derive adjoint
equations. We also give a classification of the $L^2$-shape gradient w.r.t. its
regularity and its potential to sustain the domain regularity along a descent
flow.
