Singular Structures Arising from Variational Problems in Materials Science

提案DMS-0401328PI: Peter Sternberg （Indiana University）标题：材料科学中由变分问题引起的奇异结构摘要：本提案主要研究在解决各种变分问题建模材料时出现的奇异结构。这些结构包括超导体模型中的金兹堡-朗道涡，薄膜分层模型中的水泡图案和二嵌段共聚物模型中的分离图案。在数学上，所有这些模型都涉及非凸变分问题的奇异摄动。将要用到的工具包括伽玛收敛的思想和对能量的第二种变化的仔细检查。这一建议涉及基于与各种物理构型相关的能量的数学研究的材料行为的调查。要研究的材料包括超导体和相关器件，用于制造目的的薄膜，以及由两种自分离分子形成的软“设计”材料二嵌段共聚物。根据材料自然构造自身以消耗尽可能少的能量的原则，人们从变分演算中引入数学工具，以便更好地理解这些材料的行为。

Proposal DMS-0401328PI: Peter Sternberg (Indiana University)Title: Singular structures arising from variational problems in materials scienceABSTRACT This proposal focuses on a study of singular structures that arise inthe solution to various variational problems modeling materials. Thesestructures include Ginzburg-Landau vortices in models for superconductors,blister patterns in models for thin film delamination and segregationpatterns in models for diblock copolymers. Mathematically, all of thesemodels involve singular perturbations of nonconvex variational problems. Thetools that will be brought to bear include ideas from Gamma convergence anda careful examination of the second variation of the energy. This proposal involves an investigation of material behavior based on amathematical study of the energyassociated with various physical configurations. The materials to be studiedinclude superconductors and related devices, thin films which have been laidon top of a substrate for manufacturing purposes, and diblockcopolymers--soft "designer" materials formed by two self-segregating speciesof molecules. Based on the principle that materials naturally configurethemselves so as to expend the least amount of energy possible, one bringsto bear mathematical tools from the calculus of variations so as to betterunderstand the behavior of these materials.