Minimal failure probability for ceramic design via shape control

We consider the probability of failure for components made of brittle mate- rials under one time application of a load as introduced by Weibull and Batdorf - Crosse and more recently studied by NASA and the STAU cooperation as an objective functional in shape optimization and prove the existence of optimal shapes in the class of shapes with a uniform cone property. The corresponding integrand of the objective functional has convexity properties that allow to derive lower-semicontinuity according to Fujii (Opt. Th. Appl. 1988). These properties require less restrictive regularity assumptions for the boundaries and state functions compared to [arXiv:1210.4954]. Thereby, the weak formulation of linear elasticity can be kept for the abstract setting for shape optimization as presented in the book by Haslinger and Maekinen.