Regularity Problems for Multiple Integrals and Interfacial Coarsening for Energy-Driven Models

能量驱动模型的多重积分和界面粗化的正则问题

• 批准号: 0401048
• 负责人: Xiaodong Yan
• 金额: $ 8.91万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401048&HistoricalAwards=false
• 关键词: Regularity; Problems; Multiple; Integrals; Interfacial

Proposal DMS-0401048PI: Xiaodong Yan, Michigan State UniversityTitle: Regularity properties for multiple integrals and interfacial coarsening for energy-driven models The proposal addresses problems in three main areas.In part one, we are concerned with regularity property ofminimizers of uniformly convex functionals. The objectiveis to generalize method used in previous work of Sverak andPI to construct counterexamples in lower target dimensions,as well as to understand regularity property of minimizersunder further assumptions on the integrand. The second partdeals with regularity property of minimizers for bothcompressible and incompressible models from nonlinear elasticity.The main feature here is the singular behavior of the integrandin the compressible case and nonlinear nonconvex constraint inthe incompressible case. Currently there is no general method todeal with regularity problems for such integral. We hope thatthe simple model problems studied in this project can help togain insight into more general cases. The third part addressesinterfacial coarsening phenomenon observed in many energy-drivenmodels from materials science. In particular, the aim is to providerigorous analysis on the value of the coarsening exponent for differentmodels. The analysis is closely related to deep and interesting issuesin calculus of variations, e.g.gamma limit of singularly perturbedvariational problems. The well developed tools from those areas provide abetter vision on the last project. The questions to be studied are motivated both by classical theory ofoptimization and contemporary issues in nonlinear elasticity and materialsscience. An outstanding open question in continuum mechanics is to determinewhat singularities might exist in surfaces which optimize a collection ofnonlinear constraints. Knowledge of this type can help in the design ofexotic materials. The study of coarsening is important for both bulk andthin film materials. Such materials are of great practical interest. Forexample, superconducting thin film devices are presently being evaluated forpossible use in high-performance microwave filters and other applications.

提案DMS-0401048PI：严晓东，密歇根州立大学标题：多重积分的正则性和能量驱动模型的界面粗化该提案主要解决三个方面的问题。在第一部分，我们讨论一致凸泛函的极小元的正则性。目的是推广Sverak和PI以前工作中使用的方法，在较低的目标维度上构造反例，并在被积函数的进一步假设下理解极小值的正则性。第二部分从非线性弹性的角度讨论了可压缩和不可压缩模型极小值的正则性，主要特征是可压缩情况下被积分项的奇异行为和不可压缩情况下非线性非凸约束的奇异行为。对于这类积分的正则性问题，目前还没有通用的处理方法。我们希望这个项目中研究的简单模型问题可以帮助我们深入了解更一般的情况。第三部分介绍了材料科学中许多能量驱动模型中观察到的界面粗化现象。具体地说，目的是对不同模型的粗化指数的值进行粗略的分析。这一分析与变分中深刻而有趣的问题密切相关，例如奇摄动变分问题的Gamma极限。来自这些领域的开发良好的工具为最后一个项目提供了更好的愿景。所要研究的问题既有经典的最优化理论，也有非线性弹性和材料科学的当代问题。连续介质力学中一个突出的悬而未决的问题是确定优化一组非线性约束的曲面中可能存在什么奇点。这类知识可以帮助设计奇异的材料。粗化研究对块体材料和薄膜材料都具有重要意义。这种材料具有很大的实用价值。例如，超导薄膜器件目前正在评估在高性能微波滤波器和其他应用中的可能性。