Optimality and Microstructures
Luc Tartar
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.
Luc Tartar
Carnegie Mellon University
Department of Mathematical Sciences
Pittsburgh, PA 15213
email: tartar@andrew.cmu.edu
