Degenerate Partial Differential Equations in Geometry

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Degenerate partial differential equations arise naturally in many subjects in mathematics, physics and engineering. However, the theory for degenerate partial differential equations is not well developed due to the complexity of the degeneracy. It is a common phenomenon that degeneracy causes a loss of derivatives for solutions. One of the central tasks is to identify optimal conditions under which classical solutions exist and possess nice properties. Such a task is reflected in all proposed problems. Our goal is to search for techniques to establish a priori estimates for perspective solutions under these optimal conditions. Breakthrough in this direction will have broad impacts to the whole field of degenerate partial differential equations and their applications.The investigator will carry out several research projects studying degenerate differential equations from Riemannian and complex geometry and general relativity. These include the study of Abreu's equations and extremal metrics on toric varieties, study of the generalized Jang equation and the Penrose inequality, investigation of the isometric embedding of closed surfaces in the 3-dimensional Euclidean space, and investigation of boundary behavior of minimal surfaces in the hyperbolic space. The main objectives are to understand the impact of the degeneracy on properties of solutions and to investigate the behavior of solutions near the sets of degeneracy. The discussion of the proposed mathematical problems will improve our understanding of more complicated degenerate partial differential equations in various applications.