SOME REMARKS ON SEPARATELY CONVEX FUNCTIONS LUC TARTAR* I want to discuss here some results concerning separately convex functions. Most of these results were obtained some time ago but only mentioned to a few specialists, and I had not taken the time to publish them before, for obvious reasons. The motivation of these studies was nonlinear elasticity, but once I had solved an academic example where quasiconvexity had been replaced by separate convexity, it was not clear to me how to get further on. I find useful to choose this subject now in order to describe the evolution of some ideas during the last fifteen years. YOUNG measures and separately convex functions In July 1978, I gave a few lectures at Heriot-Watt University in Edinburgh, where I described some of my ideas about how to use the compensated compact- ness method, developed in collaboration with MURAT, for attacking some difficult problems in Continuum Mechanics which interested me. This set of lectures was part of a program organized by Robin KNOPS and John BALL and I am thankful to them for having given me the opportunity to describe some of my ideas which otherwise would not have appeared in print at that time. Knowing my habit of rarely writing down the notes corresponding to my lectures, John BALL had asked one of his students, Bernard DACOROGNA, to write these notes in my place [T]. As I lectured in parallel with Ron DiPERNA it also gave me the opportunity to discuss with him my ideas of a new attack on hyperbolic systems. I had already mentioned these ideas to Ron, but not described to him all the details of what I knew at the time, and it took a few years before he could apply successfully my method to hyperbolic systems of two equations. The question of nonlinear elasticity was in my mind, even if not mentioned directly. However, my approach was quite different from that of John BALL [B], which extended the work of MORREY and was based on studying the sequentially weakly lower semi-continuous functionals, the so-called quasi convex functions. In- stead, I wanted to work with the equilibrium equations and try to identify which constitutive relations relating the stress to the strain were stable with respect to weak convergence. For that study, I advocated the old mathematical tool of Young measures together with the new tool of compensated compactness which gave con- straints that these Young measures must satisfy. Not unrelated to this approach was my desire to develop a tool which could also be used for hyperbolic systems, because I wanted to understand the evolution of elastic bodies, and I was not en- tirely satisfied with the approach of minimizing energy functionals. Of course, my program was too ambitious as even now not so much has been done in that di- rection, but I had found an (academic) example where the Young measures could *Camegie-Mellon University D. Kinderlehrer et al. (eds.), Microstructure and Phase Transition © Springer-Verlag New York, Inc. 1993