Qualitative Properties of Nonlinear Differential and Integral Equations or Systems

题目：非线性微分和积分方程或系统的定性性质[j]：李聪明（科罗拉多大学博尔德分校）摘要该项目将研究非线性积分和微分方程或系统解的定性性质，特别是那些来自物理科学或微分几何的解，这些物理或几何背景提供了很强的直觉。这种类型的研究是理解许多物理和几何问题的基础。自然现象的许多方面都是由支配它们的自然法则相互联系的，这些关系通常用微分方程或积分方程来数学描述。研究这些方程对于理解相关现象是非常重要的。通常情况下，要以足够的精度计算求解微分方程，最有效和最经济的方法是利用方程解的性质，然后开发相应的算法。除了在应用科学中非常有用外，对各种类型方程的解的各种结构和性质的研究总是会导致新的研究努力。特别是，PI将继续发展一类非常强大的技术，即微分和积分方程或系统的移动平面方法，各种基本方程组的解的分类和量化，奇异解的局部渐近对称性和解的先验估计的研究。PI还将继续与W. Chen在寻找具有规定高斯曲率或标量曲率的共形度量的几何问题上的联合工作。PI还参与了对某些抛物系统的研究，这些系统是由科学的原始分支产生的。PI将研究的中心问题是基本的Hardy-Littlewood-Sobolev不等式，包括它们的加权版本。这些不等式是Sobolev空间研究中最重要的组成部分，在非线性椭圆型和抛物型偏微分方程的研究中极为重要。第一个任务是对与这些不等式相关的泛函的所有正则临界点进行分类。二是研究奇异临界点。进一步发展以微分和积分形式移动平面的方法，对于在上述两项任务中取得进展是必不可少的，而且本身也很有趣。

DMS-0401174Title: Qualitative properties of nonlinear differential andintegral equations or systemsPI: Congming Li (University of Colorado, Boulder)ABSTRACTThe PI will study qualitative properties of solutions for nonlinear integraland differential equations or systems, especially those arising from physicalsciences or from differential geometry where the physical or geometricalbackground provides very strong intuitions. Research of this type isfundamental in the understanding of many physical and geometrical problems.Many aspects of natural phenomena are related to each other by the naturallaws governing them and very often these relations are mathematicallydescribed by differential or integral equations. The study of these equationsis very important in understanding the related phenomena. It is often the casethat to solve differential equations computationally with sufficient accuracy,the most effective and economical way is to exploit the properties ofsolutions of the equations and then to develop algorithms in accordance.Besides being very useful in applied science, the study of various kinds ofstructures and properties of solutions to various types of equationsinvariably leads to new research endeavors.In particular, the PI's will continue the development of a class of verypowerful techniques, namely the Method of Moving Planes for both differentialand integral equations or systems, to the calssification and quantization ofsolutions to various fundamental systems of equations, to the study of localasymptotic symmetry of singular solutions and a priori estimates of solutions.The PI will also continue the joint work with W. Chen on the geometric problemof finding a conformal metric with a prescribed Gaussian or scalar curvature.The PI is also involved in the study of certain parabolic systems arising formother branches of sciences. The center problem the PI will study is thefundamental Hardy-Littlewood-Sobolev inequalities including the weightedversion of them. These inequalities are the most important ingredients in thestudy of Sobolev spaces and are extremely important in the study of nonlinearelliptic and parabolic partial differential equations. The first task will bethe classification of all regular critical points of the functional associatedwith these inequalities. The second is to study the singular critical points.Further developing the method of moving planes in both differential andintegral forms is essential for making progress in the above two tasks and isinteresting in its own.