Qualitative Properties of Nonlinear Elliptic and Parabolic Equations

摘要提出研究非线性椭圆型和抛物型偏微分方程的定性性质，如解的存在性、唯一性、稳定性、对称性、先验估计和渐近性。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和解的优先估计。PI还将继续与w共同开展工作。在几何问题上寻找一个共形度量与规定的高斯或标量曲率。PI还参与研究从其他科学分支产生的某些抛物线系统。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在数学上加以描述。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，为了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。

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.