Mathematical Sciences: Calculus of Variations, Material Microstructure, and Stochastic Evolution Problems

计划了三个不同但松散相关的研究项目：(A)研究等参数约束问题的Lavrentiev现象及其与非线性弹性中空化效应的关系；(B)研究具有层状微结构的材料和具有负毛细作用的材料；以及(C)向列相液晶中的随机双曲方程和随机控制。每一部分都代表着研究人员在先前工作中开发的数学工具的进一步发展，以及新研究课题的引入。这三个项目的动机是来自连续介质力学的问题。其目的是了解各种新材料的物理学，并发展有助于解释、预测和控制它们的力学行为的数学。

Three distinct but loosely related research projects are planned: (a) A study of the Lavrentiev phenomenon for isoperimetrically constrained problems and its relationship with cavitation effects in nonlinear elasticity, (b) A study of materials with layered microstructure and materials with negative capillarity, and (c) Stochastic hyperbolic equations and stochastic control in nematic liquid crystals. Each part represents a further development of the mathematical tools developed in the investigator's previous work, as well as the introduction of new research topics. The three projects are motivated by problems from continuum mechanics. The objective is to understand the physics of an assortment of novel materials and to develop mathematics which will help to explain, predict, and control their mechanical behavior.