"Optimality and Microstructures''

ABSTRACT: In joint work with Francois MURAT in the early 70s, we studied an approach to Optimal Design questions where the study of microstructures and their effective properties was crucial. For these questions, we developed the theory of Homogenization (H-convergence) and the theory of Compensated Compactness, and I adapted the classical tool of parametrized measures used in control theory (later called YOUNG measures) for describing the implications of the Compensated Compactness Method for problems with microstructures. Later, I developed the theory of H-measures for improving these questions. The lectures will describe in what situations one should expect the apparition of microstructures, how to estimate the corresponding effective properties and why YOUNG measures cannot answer these questions in general, and how the question of understanding local optimality of microstructures is different from the application of the Gamma-convergence approach due to Ennio DE GIORGI.