Mathematical Sciences: Calculus of Variations/Control, and Applications to Material Science

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DMS-9500915 Mizel This is a proposal for a research project in mathematics. Part I of this proposal involves work on the foundations of the calculus of variations and control theory. Part II involves work on simplified models for materials possessing higher order and nontrivial microstructure, as well as work involving control problems for matterials which are in the liquid crystal state. In each part, the proposed research represents, in part, a further development of previously accomplished work, and , in part, the introduction of new research topics. In part I, renewed interest in a gap phenomenon discovered by Lavrentiev in 1926 has come about because of the potential implications for nonlinear elasticity and related subjects. The recently gained insights into this phenomenon in first order one-dimensional Lagrange problems are now being supplemented by the study of higher order problems in one-dimension, by the study of first order problems in higher dimensions, and by the study of isoperimetrically constrained problems in higher dimensions. In part II, a homogenization method for examining a simplified model of a second order nonlinear material leads to a characterization and minimization problem involving Young measures which is currently being investigated, and a study of the implications of polyconvexity for initation of fracture in isotropic materials is under way..