Contemporary methods in calculus of variations and differential equations with applications to materials science

这个项目的中心主题是现代应用数学在材料、生物科学和力学中的应用。主要的研究领域有：1)生物膜中的相分离和结构域形成。许多重要的生物学过程，如病毒萌发和免疫反应，被认为与膜筏有关。2)弹性材料中缺陷的运动。位错是晶体材料中最常见的缺陷，其流动性决定了大多数金属的塑性行为和延展性。3)外延生长的演化问题。了解薄膜在衬底上的外延沉积过程在半导体电子和光电子器件的制造中具有重要意义，例如量子阱激光器。作为该项目的一部分，研究人员继续通过几项整合研究和教育的活动对学生进行培训，包括：专题课程、暑期学校、工作组、课程设计研讨会和职业发展工作坊。三个主要主题领域的具体目标是：1)膜和囊泡的相变；材料不稳定性模型的一个共同特征是，它们经常表现出几个尺度，有效行为通过渐近分析得到解释。这里的目标是解析地研究Andelman、Kawasaki、Kawakatsu和Taniguchi提出的描述单层膜在平面内相分离过程中的形状变形的变分模型。2)晶体材料中螺位错的运动。这项工作的主要目的是利用Filippov发展的微分夹杂理论来研究Cermelli和Gurtin的有限数量螺旋Volterra位错动力学模型。3)外延生长的演化问题。在这里，研究人员得到了各向异性外延应变薄膜表面在三维应力和表面质量输运驱动下的非线性形态演化和岛屿形成的严格解析结果。

The central theme of this project is the application of modern applied mathematics to materials and biological sciences and mechanics. Thematic areas of focus are: 1) Phase separation and domain formation in biomembranes. Many important biological processes such as virus budding and immune responses are believed to be linked to membrane rafts. 2) Motion of defects in elastic materials. Dislocations are the most common defects in crystalline materials, and their mobility is responsible for the plastic behavior and the ductility of most metals. 3) Evolution problems for epitaxial growth. Understanding the epitaxial deposition process of a thin film onto a substrate is of central importance in the manufacturing of semiconductor electronic and optoelectronic devices, such as quantum well lasers. As part of this project the investigator continues training of students through several activities that integrate research and education, including: topics courses, summer schools, working groups, course design workshops, and career-building workshops.Specific objectives in the three main thematic areas of focus are: 1) Phase Transitions of Membranes and Vesicles; A common feature of models issuing from materials instabilities is that they often exhibit several scales, and effective behavior is interpreted via asymptotic analysis. The goal here is to study analytically a variational model introduced by Andelman, Kawasaki, Kawakatsu, and Taniguchi to describe the shape deformations of unilamellar membranes undergoing an in-plane phase separation. 2) Motion of screw dislocations in crystalline materials. The main objective of this work is to use the theory of differential inclusions developed by Filippov to study a model of Cermelli and Gurtin for the kinetics of a finite number of screw Volterra dislocations. 3) Evolution problems for epitaxial growth. Here the investigator obtains rigorous analytical results for the nonlinear morphologic evolution of the surfaces of anisotropic epitaxially strained films and the formation of islands, driven by stress and surface mass transport in three dimensions.