Regularity of Degenerate Partial Differential Equations and Applications

偏微分方程（PDEs）是科学用来模拟各种过程的数学语言，如热流、波传播、人类和动物的生物功能、金融市场、天体动力学、结构工程以及许多其他过程。大多数科学原理和理论都可以用微分方程优雅地表述出来。正因为如此，自从牛顿和莱布尼茨为微积分奠定基础以来，偏微分方程就一直被广泛研究，今天人们对所谓的偏微分方程经典理论有了很好的理解。然而，许多应用并不符合经典模型的限制，因为控制其性质的方程在某些方面是“简并的”。

Partial Differential Equations (PDEs) is the mathematical language science uses to model a wide variety of processes like heat flow, wave propagation, biological functions in humans and animals, financial markets, celestial dynamics, and the engineering of structures, among many other processes. Most scientific principles and theories can be elegantly stated using differential equations. For this reason PDEs have been extensively studied ever since Newton and Leibniz set the foundations for differential Calculus, and today there is a good understanding of what is now called classical theory of PDEs. However, many applications do not conform to the restrictions of classical models, as the equations governing their nature are ''degenerate'' in some aspects. Consequently, the objective of this proposal is to investigate key properties of solutions to PDEs with different types of degeneracies. The problems considered lie on the fringe of known theory, but escape classical techniques for their treatment. The importance of exploring these frontiers resides on the deeper insights gained for the general theory of PDEs and on the newly developed techniques that will offer potential for further applications.