Critical points of variational integrals

变分积分的临界点阎晓东本课题主要研究变分微积分中多重积分临界点的规律性；特别是一些由非线性弹性引起的积分。还有一些经典分析中众所周知的结果的一般化问题。一个课题是研究保证临界点部分正则性的积项的条件。特别地，我们将研究某些非线性弹性模型系统的规律性。这将包括可压缩和不可压缩的材料。另一个课题是研究某些非线性积分方程组解的规律性和定性性质。这些系统与Hardy-Littlewood-Sobolev不等式中出现的系统密切相关。本课题研究的问题是分析和变分法中的重要技术成果。任何结果都将引起相当大的兴趣，因为在过去的二十年里，有很多关于这些和相关主题的研究。非线性弹性问题可能导致对奇异材料的行为的深入了解，而其他问题可能对某些几何问题很重要。

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.