Dynamic Free Boundary Problems

动态自由边界问题处理的是在未知先验演化区域内求解偏微分方程的问题。一个例子是水冻结成冰的问题，其中冰和水的界面是由水区域的热方程的解（温度分布）决定的。这个项目关注的是那些解决方案几乎没有规律性的问题。例如，当水结冰时，产生的冰晶预计会在其边界上形成分形结构。另一个例子是肿瘤生长，由于寻找高营养区域，肿瘤的边界变得不规则。该项目旨在更好地理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框架。本项目为研究生提供研究训练机会。低维结构的存在在物理文献中无处不在，要么作为一个领域的边界，要么作为一个进化的单一部分。非线性偏微分方程中的许多问题，当它们与移动界面相结合时，即使在看似简单的设置中，也会带来重大挑战。除了非线性之外，问题的难点还在于它的非局域性，即解的行为取决于自由边界的整体几何形状。动态自由边界问题由于缺乏自由边界的先验规律性而面临额外的困难。通过研究提出的问题，首席研究员（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的论点。该奖项反映了美国国家科学基金会的法定使命，并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.