Qualitative Properties of Nonlinear Elliptic and Parabolic Equations

DMS-9970530李聪明研究非线性椭圆和抛物型偏微分方程解的存在性、唯一性、稳定性、对称性、先验估计和渐近性等定性性质。这种类型的研究是理解许多物理问题的基础。PI将继续发展一类非常强大的技术，即移动平面方法，以研究奇异解的局部渐近对称性和解的优先估计。PI还将继续与W. Chen关于求一个具有近似高斯曲率或标量曲率的共形度量的几何问题。PI也参与了其他科学分支中某些抛物系统的研究。自然现象的许多方面都是由控制它们的自然规律相互关联的，而且这些关系通常用微分方程来数学描述。对这些方程的研究对于理解相关现象是非常重要的，通常情况下，要用计算的方法来求解微分方程并达到足够的精度，最有效和最经济的方法是利用方程解的性质，然后根据这些性质发展相应的算法。除了在应用科学中非常有用之外，对各种类型方程的解的各种结构和性质的研究也导致了新的研究努力。

DMS-9970530Congming LiABSTRACTThe PI proposes to study qualitative properties of nonlinearelliptic and parabolic partial differential equations (PDEs), suchas the existence, uniqueness, stability, symmetry, a priori estimates,and asymptotic behavior of solutions. This type of research isfundamental to the understanding of many physical problems governed byPDEs. The PI's will continue the development of a class of verypowerful techniques, namely the Method of Moving Planes, to the studyof local asymptotic symmetry of singular solutions and a prioriestimates of solutions. The PI will also continue the joint work withW. Chen on the geometric problem of finding a conformal metric with aprescribed Gaussian or scalar curvature. The PI is also involvedin the study of certain parabolic systems arising form other branchesof sciences.Many aspects of natural phenomena are related to each other by thenatural laws governing them and very often these relations aremathematically described by differential equations. The study of theseequations is very important in understanding the related phenomena.It is often the case that to solve differential equations computationally with sufficient accuracy, the most effective andeconomical way is to exploit the properties of solutions of the equations and then to develop algorithms in accordance}. Besides beingvery useful in applied science, the study of various kinds ofstructures and properties of solutions to various types of equationsinvariably leads to new research endeavors.