Regularity of Degenerate Partial Differential Equations and Applications

偏微分方程（PDE）是一种数学语言，科学用来模拟各种各样的过程，如热流，波传播，人类和动物的生物功能，金融市场，天体动力学和结构工程，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。由于这个原因，自从牛顿和莱布尼茨为微分学奠定基础以来，偏微分方程就被广泛研究，今天人们对偏微分方程的经典理论有了很好的理解。然而，许多应用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“退化”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.