Degenerate Partial Differential Equations in Geometry

退化偏微分方程在数学、物理和工程学的许多学科中都是自然产生的。然而，由于退化偏微分方程的复杂性，退化偏微分方程的理论还没有得到很好的发展。简并性导致解的导数损失是一个普遍现象。其中一个中心任务是确定经典解存在并具有良好性质的最优条件。所有提出的问题都反映了这一任务。我们的目标是寻找技术，建立先验估计的角度解决方案在这些最佳条件下。这一方向的突破将对整个退化偏微分方程及其应用领域产生广泛的影响。研究者将开展多个研究项目，从黎曼几何和复几何以及广义相对论的角度研究退化微分方程。其中包括研究Abreu方程和环面簇的极值度量，研究广义Jang方程和Penrose不等式，研究三维欧氏空间中闭曲面的等距嵌入，以及研究双曲空间中极小曲面的边界行为。主要目的是了解退化对解的性质的影响，并研究解在退化集附近的行为。对这些数学问题的讨论将有助于提高我们对更复杂的退化偏微分方程在各种应用中的理解。

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.