Qualitative Properties of Nonlinear Differential and Integral Equations or Systems

DMS-0401174题目：非线性微分和积分方程或系统的定性性质PI：李聪明（科罗拉多大学，博尔德）摘要PI将研究非线性积分和微分方程或系统的解的定性性质，特别是那些来自物理科学或微分几何的物理或几何背景提供了非常强的直觉。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规律相互联系的，而且这些关系常常用微分或积分方程来描述。这些方程的研究对于理解相关现象具有重要意义。为了在计算上达到足够的精度，最有效和最经济的方法是利用方程解的性质，然后发展相应的算法。除了在应用科学中非常有用外，对各种类型方程解的各种结构和性质的研究也在不断地导致新的研究努力。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，以分类和量化各种基本方程系统的解，研究奇异解的局部渐近对称性和解的先验估计，PI也将继续与W.他的主要研究方向是寻找具有高斯曲率或纯量曲率的共形度量的几何问题。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，在非线性椭圆和抛物型偏微分方程的研究中也是极其重要的。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，进一步发展微分和积分形式的动平面法是在上述两项工作中取得进展的必要条件，它本身也是很有意义的。

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.