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

9500915毫泽尔这是一个数学研究项目的提案。这项建议的第一部分涉及变分和控制理论的基础工作。第二部分涉及具有高阶和非平凡微结构的材料的简化模型的工作，以及涉及液晶状态的材料的控制问题的工作。在每一部分中，拟议的研究部分是对以前完成的工作的进一步发展，部分是引入新的研究课题。在第一部分中，拉夫伦蒂耶夫在1926年发现的一种缺口现象重新引起了人们的兴趣，因为它对非线性弹性和相关学科具有潜在的影响。最近在一阶一维Lagrange问题中对这一现象的深入研究现在被一维高阶问题的研究、高维一阶问题的研究以及高维等周约束问题的研究所补充。在第二部分中，一种用于检验二阶非线性材料简化模型的齐次化方法导致了涉及Young测度的表征和最小化问题，该问题目前正在研究中，并且正在研究多凸性对各向同性材料中的起裂的影响。

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..