Singular Structures Arising from Variational Problems in Materials Science

提案DMS-0401328 PI：Peter斯滕贝格（印第安纳州大学）标题：变分问题引起的奇异结构 在材料科学中的应用摘要这个提议集中在奇异结构的研究上，这些奇异结构是在对材料建模的各种变分问题的解决方案中出现的。这些结构包括超导体模型中的Ginzburg-Landau涡旋，薄膜分层模型中的气泡图案和二嵌段共聚物模型中的偏析图案。在数学上，所有这些模型都涉及非凸变分问题的奇异摄动。这些工具将带来承担包括伽玛收敛和仔细检查的第二变化的能量的想法。 这个建议涉及到对材料行为的研究，其基础是对与各种物理结构相关的能量进行数学研究。待研究的材料包括超导体和相关器件，为制造目的而放置在衬底顶部的薄膜，以及二嵌段共聚物--由两种自分离分子形成的软“设计师”材料。基于材料自然地配置自己以便消耗尽可能少的能量的原理，人们从变分法中引入数学工具，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.