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

DMS-9500915米泽尔 这是一个数学研究项目的建议书。本提案的第一部分涉及变分法和控制理论的基础工作。 第二部分涉及的工作简化模型的材料具有更高的秩序和非平凡的微观结构，以及工作涉及控制问题的材料是在液晶状态。在每一部分中，拟议的研究部分代表了以前完成的工作的进一步发展，部分代表了新的研究课题的引入。 在第一部分中，重新对Lavrentiev于1926年发现的间隙现象产生了兴趣，因为它对非线性弹性和相关学科具有潜在的影响。最近获得的洞察这一现象，在一阶一维拉格朗日问题，现在正在补充的研究，在一维的高阶问题，在高维的一阶问题的研究，并在高维的isoperimetrically约束问题的研究。在第二部分中，一个均匀化的方法，用于检查一个简化的模型的二阶非线性材料导致的表征和最小化问题，涉及杨措施，目前正在调查，和研究的影响polyconvexity在各向同性材料中的断裂起始正在进行中。

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..