Qualitative Properties of Nonlinear Differential and Integral Equations or Systems

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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.