Asymptotic Behavior of Singular Solutions to Some Geometric PDEs
Professor Zheng-chao Han
Department of Mathematics
Rutgers University
Abstract: Many geometric problems or their resolution involve the study of singular solutions to some relevant geometric equations, even if the original problems themselves do not directly involve singular solutions, as singular solutions may arise in the analysis of limits of regular solutions. One basic question is whether there is a precise description of the behavior of the singular solution near its singular set. The study of this kind of questions also leads to the study of singular solutions defined on the entire Euclidean space with some lower dimensional Euclidean space removed --- this arises through zooming in near the singular set, a typical procedure for such problems. In particular, whether such limiting solutions are unique, or are from a finite parameter of family of explicit or known solutions? There are no general answers to such general questions, but, for certain geometric PDEs, exploitation of covariance properties of certain geometric PDEs has led to some useful answers to these questions. I will use a few examples to illustrate how such questions are formulated and analyzed.
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