We consider the existence of optimal shapes in the context of the
thermomechanical system of partial differential equations (PDE) using the
recent approach based on elliptic regularity theory. We give an extended and
improved definition of the set of admissible shapes based on a class of
sufficiently differentiable deformation maps applied to a baseline shape. The
obtained set of admissible shapes again allows one to prove a uniform Schauder
estimate for the elasticity PDE. In order to deal with thermal stress, a
related uniform Schauder estimate is also given for the heat equation. Special
emphasis is put on Robin boundary conditions, which are motivated from
convective heat transfer. It is shown that these thermal Schauder estimates can
serve as an input to the Schauder estimates for the elasticity equation. This
is needed to prove the compactness of the (suitably extended) solutions of the
entire PDE system in some state space that carries a c2-H\"older topology for
the temperature field and a C3-H\"older topology for the displacement. From
this one obtains he property of graph compactness, which is the essential tool
in an proof of the existence of optimal shapes. Due to the topologies employed,
the method works for objective functionals that depend on the displacement and
its derivatives up to third order and on the temperature field and its
derivatives up to second order. This general result in shape optimization is
then applied to the problem of optimal reliability, i.e. the problem of finding
shapes that have minimal failure probability under cyclic thermomechanical
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