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

题目：能量驱动模型的多重积分和界面粗化的正则性。该提案主要解决三个方面的问题。在第一部分中，我们讨论了一致凸泛函的极小值的正则性。目的是推广Sverak和pi之前的工作中使用的方法，以在较低的目标维上构造反例，并在进一步的被积函数假设下理解最小值的正则性。第二部分从非线性弹性角度讨论了可压缩模型和不可压缩模型的极小值的正则性。其主要特征是可压缩情况下被积函数的奇异性和不可压缩情况下的非线性非凸约束。对于这类积分的正则性问题，目前还没有一般的处理方法。我们希望在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。第三部分讨论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。特别地，目的是对不同模型的粗化指数值提供严谨的分析。该分析与变分学中一些深刻而有趣的问题密切相关，如奇摄动变分问题的极限。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远景。要研究的问题是由经典的优化理论和非线性弹性和材料科学的当代问题的动机。连续介质力学中一个突出的开放性问题是确定在优化非线性约束集合的曲面上可能存在哪些奇点。这方面的知识有助于异域材料的设计。粗化研究对块状材料和薄膜材料都具有重要意义。这些材料具有很大的实用价值。例如，超导薄膜器件目前正在被评估用于高性能微波滤波器和其他应用的可能性。

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.