9704813 Grabovsky This proposal aims to investigate three problems related to homogenization, composite materials and their effective properties. The first problem will combine homogenization theory, bounds on the effective moduli of composites and Young measure techniques in order to study the relaxation of the ill-posed optimization problems with weakly discontinuous objective functionals. The method yields an answer for a class of quadratic functionals. The second project will seek out the exact relations for the effective behavior of polycrystalline composites using a new technique that combines PDE theory and variational principles. The method is successful in at least one such problem. The third project will study the critical behavior of high contrast composites near the percolation threshold by utilizing the analogy between the phase transitions in Ising ferromagnets and the random resistor networks near the percolation threshold on the one hand, and recent results on the high contrast continuum composites away from the critical point on the other. Composite materials are media that look homogeneous but in fact have complex structure when viewed under a microscope. These materials are finding their way into our everyday lives in objects such as skis, golf clubs, automobiles, aircraft, computers, cement, construction components of buildings and bridges, sensors and many many more. The idea of composite materials is to combine beneficial properties of its simple but imperfect constituents. The difficulty here is that the properties of composites (called effective properties) depend not only on what you mix but on how you mix it. In other words, they depend on the geometry of what you see under a microscope (called microgeometry). Therefore the goal is to specify which microgeometries produce the best composites provided you use the same materials in prescribed proportions. In general this is an ill-posed problem. That is, it may not be solved in its raw form by the most powerful computers. The first part of this proposal aims to reformulate the problem into a form suitable for solution on a computer (into a well-posed problem). The second part of this proposal will seek out and investigate the very special cases when some effective properties of polycrystalline composites do not depend on the microgeometry. This work may have applications for the design of underwater sensors based on piezoelectric polycrystals. The last part of this proposal constitutes a long term project on studying "nature's composites" like porous rocks or sea ice (consisting of ice with brine inclusions). The problem here is that the two constituents comprising the composite have extremely different properties (like air and rock, or brine and ice). Modeling the properties of such composites presents a formidable mathematical problem. The progress here would enhance our understanding and give us more control of a diverse array of problems ranging from extraction of oil from porous rocks to global climate change (that depends to a great extent on what is happening to Arctic and Antarctic sea ice).
