Free Boundary Problems in Partial Differential Equations

在自由边界问题中，人们寻求确定PDE系统在一个域内的解，以及域本身的边界；自然地，将未知边界（“自由边界”）与未知解连接起来的补充条件被施加。这种情况源于作为模型基础的物理定律。当研究者试图解决特殊的自由边界问题时，产生了一些一般理论。例如，变分不等式理论是从弹性接触问题的研究和凝固问题发展而来的。本项目正在进行中，主要研究两个方面：(1)裂纹扩展，其中PDE是双调和方程的解，沿裂纹的牵引力为零，裂纹尖端按照断裂力学规律的I型或II型及时扩展；(2)一个由两个反应扩散方程组成的系统，描述肿瘤细胞的压力和肿瘤内的营养物质浓度，边界处有守恒定律。目标是推导出存在性、唯一性、稳定性和定性。这些是目前理论未涵盖的新型自由边界问题。第一个项目的结果将有助于更好地理解裂纹如何在弹性材料中传播，这一问题在电路板布线和老化的飞机面板中变得越来越重要。第二个项目涉及数学生物学家近年来开发的简单肿瘤模型。我们的研究将提供一些关于肿瘤的大小和形状如何随时间演变的理解。

In free boundary problems one seeks to determine a solution of PDE system in a domain, as well as the boundary of the domain itself; naturally a supplementary condition is imposed which connects the unknown boundary (the "free boundary") to the unknown solution. This condition arises from the physical laws which underlie the model. Some general theories developed as researchers tried to resolve special free boundary problems. For instance, the theory of variational inequalities evolved from the study of contact problems in elasticity and from solidification problems. The present project is ongoing work, primarily in two areas: (1) Propagation of cracks, where the PDE is a solution of the biharmonic equation, with zero traction along the crack, and the crack'9s tip propagate in time in accordance with mode I or mode II of fracture mechanic laws; (2) A system of two reaction-diffusion equations for the pressure of tumor cells and nutrient concentration within an evolving tumor, with a conservation law at the boundary. The objective is to derive existence, uniqueness, stability and qualitative properties. These are new types of free boundary problems not covered by present theories.The results of the first project will lead to better understanding of how cracks propagate in elastic material, a problem which is becoming increasingly important in the wiring of circuit boards and in aging airplane panels, for example. The second project deals with simple tumor models that mathematical biologists have developed in recent years. Our research will provide some understanding of how the size and shape of the tumor will evolve in time.