Critical points of variational integrals

变分积分的临界点建议研究摘要严晓东本项目研究变分学中多重积分临界点的规律性，特别是非线性弹性力学中的一些被积函数。同时也推广了经典分析中的一些著名结果。一个项目是研究保证临界点部分正则的被积函数的条件。特别是在非线性弹性模型的某些系统的规律性将被研究。这将包括可压缩和不可压缩材料。另一个项目是研究某些非线性积分方程组解的正则性和定性性质。这些系统与Hardy-Littlewood-Sobolev不等式中的系统密切相关，所研究的问题是变分分析中的重要技术结果。任何结果都将是相当大的兴趣，因为在过去的二十年里，对这些问题和相关问题进行了大量的研究。非线性弹性力学中的问题可能导致对奇异材料行为的深入了解，而其他问题可能对几何中的某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.