NSF Award
2305038
This project involves nonlinear elliptic partial differential equations that arise in geometry, physics, and engineering. The equations studied in this project exhibit a certain degree of degeneracy or singularity. Degenerate elliptic equations demonstrate significant differences from uniformly elliptic equations, an important class of nondegenerate counterparts. Solutions of degenerate elliptic equations have much worse properties than those of uniformly elliptic equations (typically, a loss of smoothness of derivatives at some level). The classical analysis of uniformly elliptic equations, emphasizing the regularity of solutions (i.e., smoothness of their derivatives), is inadequate for studying degenerate elliptic equations. The theme of the project is the analysis of solutions to these equations near sets where degeneracy or singularity occurs. One of the main tasks is to study the properties of solutions in new formats and to identify optimal conditions for the existence and regularity of solutions. Such a task is reflected in all problems in this project. The central part of this project is dedicated to the development of new methods and techniques in studying the properties of solutions of degenerate and singular elliptic differential equations. In addition to pursuing fundamental problems in geometry and physics, this project also contributes to establishing a general framework to study several important classes of degenerate elliptic differential equations and advancing our knowledge of these equations. Student training and interdisciplinary collaborations are important aspects of this project. Three graduate students in mathematics and one graduate student in electrical engineering participate in this project. <br/><br/>The project will study degenerate or singular elliptic partial differential equations. Due to the complexity and diversity of the types of degeneracy, it is impossible to develop a general theory for degenerate elliptic equations. The project focuses on three classes of degenerate elliptic equations that appear frequently in geometry and physics. The first class is the uniformly degenerate elliptic equations. Many important geometric problems are reduced to this class of equations. These equations are defined on compact manifolds with boundaries and degenerate only on boundaries, with a uniform rate of degeneracy in terms of the distance to the boundary. The second class of equations consists of elliptic equations that are singular at isolated points, with a uniform rate of singularity in terms of the distance to the singular point. Asymptotic expansions demonstrate that solutions to these two classes of equations are singular with specific singular factors. The primary objective of the research on these equations is to study the impact of these singular factors on the existence and regularity of solutions. The third class of equations consists of elliptic equations and systems of elliptic equations that are singular in a set of codimension 2. This class of equations appears in the study of radially symmetric solutions to the Einstein equation, near the extreme Kerr solution. The research on this class of equations will study the asymptotic behaviors of solutions near punctures and to prove the mass and angular momentum inequality. Completely different methods are required to study these three classes of equations, although they differ only in the dimension of the sets where degeneracy occurs.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
