Regularity of Degenerate Partial Differential Equations and Applications

偏微分方程(PDE)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、天体动力学和结构工程等许多过程。大多数科学原理和理论都可以用微分方程式优雅地表述出来。由于这个原因，自从牛顿和莱布尼茨建立了微积分的基础以来，偏微分方程组就得到了广泛的研究，今天人们对偏微分方程组的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“退化的”。 因此，本提案的目的是研究具有不同类型退化的偏微分方程解的关键性质。所考虑的问题存在于已知理论的边缘，但没有解决这些问题的经典技术。探索这些前沿的重要性在于对偏微分方程一般理论获得的更深层次的见解，以及将为进一步应用提供潜力的新开发的技术。

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.