Shape problems and differential inequalities

In this paper we present numerous problems of shape optimal design that arise in structural mechanics, fluid flows, and other areas of engineering and applied science, for systems described by differential inequalities. Several methods for characterizing domains are identified. A variational formulation of the state problem, and of the derivative of the cost function, is presented throughout, in preparation for the development of numerical methods of solution. Examples are used to clearly illustrate the essential elements of shape optimal design problems.