Critical points of variational integrals

变分积分的临界点建议研究摘要严晓东这个项目是研究变分中重积分临界点的规律性，特别是一些由非线性弹性引起的被积分点的规律。还有一些推广经典分析中著名结果的问题。一个项目是研究被积函数上保证临界点的某些局部正则性的条件。特别地，将研究作为非线性弹性模型的某些系统的正则性。这将包括可压缩和不可压缩的材料。另一个项目是研究某些非线性积分方程解的正则性和定性性质。这些系统与Hardy-Littlewood-Sobolv不等式中的系统密切相关。本项目要研究的问题是分析和变分法中的重要技术成果。任何结果都会引起相当大的兴趣，因为在过去的二十年里，关于这些和相关的主题已经有了很多研究。非线性弹性问题可能导致对奇异材料行为的洞察，而其他问题可能对几何中的一些问题具有重要意义。

Critical Points of Variational IntegralsAbstract of Proposed Research Xiaodong YanThis project is to study the regularity of critical points of multiple integrals in the calculus of variations; especially some of the integrands that arise from nonlinear elasticity. Also some questions that generalize well-known results in classical analysis. One project is to investigate conditions on the integrands that guarantee certain partial regularity of critical points. In particular the regularity of certain systems that are models in nonlinear elasticity will be studied. This will include both compressible and incompressible materials. Another project is to study the regularity and qualitative properties of the solutions of certain systems of nonlinear integral equations. These systems are closely related to the systems that arise in the Hardy-Littlewood-Sobolev inequality.The questions to be studied in this project are important technical results in analysis and the calculus of variations. Any results would be of considerable interest as there has been much research on these, and related, subjects during the last twenty years. The problems in nonlinear elasticity may lead to insights into the behavior of exotic materials while the other problems may be of importance for some questions in geometry.