Dynamic Free Boundary Problems

动态自由边界问题是求解区域内的偏微分方程，其演化过程是未知的。一个例子是水冻结成冰的问题，其中冰和水的界面由水区域的热方程的解（温度分布）确定。这个项目的重点是那些解决方案几乎没有规律性的问题。例如，当水冻结成冰时，所产生的冰晶预计会在其边界上形成类似分形的结构。另一个例子是肿瘤生长，由于寻找高营养区域，肿瘤的边界发展成不规则的。该项目旨在更好地了解这些问题的解决方案的性质，并为开发精确的数值模拟提供一个框架。该项目为研究生提供了研究培训的机会。低维结构的存在在物理文献中无处不在，无论是作为域的边界还是演化的奇异部分。非线性偏微分方程中的许多问题，否则很好地理解时，他们与一个移动的界面耦合，即使在看似简单的设置提出了重大的挑战。除了非线性，问题的困难在于它的非局部性，在这个意义上说，解的行为取决于自由边界的整体几何。动态自由边界问题面临着一个额外的困难，来自于缺乏一个先验的规则性的自由边界。通过研究提出的问题，主要研究者（PI）的目的是制定一般的方法来处理这些困难。该项目的工作使用几何分析以及变分和概率论证。该奖项反映了NSF的法定使命，并且通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Dynamic free boundary problems deal with solving partial differential equations in domain whose evolution is not known a priori. One example is the problem of water freezing into ice, where the interface of ice and water is determined by the solution of the heat equation in the water region (the distribution of temperature). This project is focused on problems whose solutions have very little regularity. For instance, when water freezes into ice, the resulting ice crystal is expected to form a fractal-like structure on its boundary. Another example is tumor growth, where the boundary of the tumor develops irregularity due to the search for regions of high nutrients. The project is aimed at providing a better understanding of the properties of the solutions to such problems and provide a framework for developing accurate numerical simulations. The project provides research training opportunities for graduate students.The presence of lower-dimensional structures is ubiquitous in the physical literature, either as a boundary of a domain or a singular part of an evolution. Many problems in nonlinear partial differential equations which are otherwise well understood pose significant challenges when they are coupled with a moving interface, even in seemingly simple settings. Besides the nonlinearity, the difficulty of the problem lies in its nonlocality, in the sense that the behavior of solutions depends on the global geometry of the free boundary. Dynamic free boundary problems face an additional difficulty coming from the lack of a priori regularity of the free boundaries. By studying the proposed problems, the Principal Investigator (PI) aims to develop general methods to deal with such difficulties. The work of the project uses geometric analysis as well as variational and probabilistic arguments.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.