Singular Structures Arising from Variational Problems in Materials Science

Indiana University, Bloomington IN

Investigators

Abstract

Proposal DMS-0401328 PI: Peter Sternberg (Indiana University) Title: Singular structures arising from variational problems in materials science ABSTRACT This proposal focuses on a study of singular structures that arise in the solution to various variational problems modeling materials. These structures include Ginzburg-Landau vortices in models for superconductors, blister patterns in models for thin film delamination and segregation patterns in models for diblock copolymers. Mathematically, all of these models involve singular perturbations of nonconvex variational problems. The tools that will be brought to bear include ideas from Gamma convergence and a careful examination of the second variation of the energy. This proposal involves an investigation of material behavior based on a mathematical study of the energy associated with various physical configurations. The materials to be studied include superconductors and related devices, thin films which have been laid on top of a substrate for manufacturing purposes, and diblock copolymers--soft "designer" materials formed by two self-segregating species of molecules. Based on the principle that materials naturally configure themselves so as to expend the least amount of energy possible, one brings to bear mathematical tools from the calculus of variations so as to better understand the behavior of these materials.