1. The first lecture gives an outline on the engineering practice of
fatigue life calculation for mechanical components under cyclic loading.
The elasticity PDE and the weak solution theory are developed, physical
failure mechanisms as Low Cycle Fatigue are reviewed. The combination
of Neuber- and Glinker shake-down, Ramberg Osgood equantion and
Coffin-Manson Basquin equation are explained with an emphasis to
mathematical properties. Stochastic failure time processes are
introduced on the basis of the first occurrence time of spatio-temporal
point processes. A specific failure time model is introduced based on
the Weibull distribution. It is explained, how the demand for optimal
reliability, i.e. the choice of a form that minimizes failure
probabilities, in a natural way leads to a problem in the field of shape
optimization.

2. The second lecture reviews the fundamentals
of shape optimization including definitions of admissible shapes, shape
compactness, the relation to the state equation and the requirement of
lower semicontinuity. These features are combined into an abstract proof
of the existence of optimal shapes. The role of regularity theory in
the existence proof of shapes with optimal reliability properties is
explained. We shortly introduce the theory of strong solutions and
boundary regularity (Schauder esimates) that allow to show the crucial
graph compactness properties needed in the abstract existence proofs.
Uniform Schauder estimates are provided and the proof of existence of
forms with minimal failure probabilities is completed.

3. In the
last lecture, an exposition of the notion of shape derivatives and shape
gradients is given. We sketch a proof, how the existence of shape
derivatives for very singular objective functionals, motivated by
failure probabilities, can actually be constructed. Implications to
optimization on the infinite dimensional manifold of shapes are given.