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

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

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.