Mathematical Sciences: Nonconvex Variational Problems and Material Instabilities CMU Proposal 06823

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DMS-9500531 Fonseca The general objective of this project is the study of variational problems in connection with material instabilities. The phenomema addressed will include phase transitions, metastable equilibrium states and the presence of dislocations and other defects in crystals, the onset of microstructure and the creation of concentrations, interaction between fracture and damage, and the behavior and defects of liquid crystals. Their study escapes the framework of classical mathematical theories, and it concentrates on very contemporary research issues in material science. The success of this program will require manipulation of recently developed mathematical tools, the introduction of new ones, and it may have a considerable impact in the advance of industry and technology, clarifying problems in material sciences and smart materials. The areas involved in this program are Continuum Mechanics, Geometric Measure Theory, Partial Differential Equations, Thermoelastodynamics, and the Calculus of Variations. Among the underlying mathematical questions, one can find constrained variational problems, evolution of phase boundaries, interfacial energies, relaxation techniques, and generalized measure-valued solutions.