Free Boundary Problems in Partial Differential Equations

在自由边界问题中，人们试图确定区域中PDE系统的解以及区域本身的边界;自然地，施加一个补充条件，将未知边界（“自由边界”）连接到未知解。 这个条件是由作为模型基础的物理定律引起的。 随着研究人员试图解决特殊的自由边界问题，一些一般理论得到了发展。 例如，变分不等式理论是从弹性接触问题和固化问题的研究发展而来的。 本课题的主要研究内容包括两个方面：（1）裂纹的扩展，其中偏微分方程是双调和方程的一个解，沿裂纹沿着方向的牵引力为零，裂纹尖端按断裂力学规律的Ⅰ型或Ⅱ型随时间扩展;（2）一个由两个反应扩散方程组成的系统，该系统描述了肿瘤细胞的压力和肿瘤内营养物质的浓度，边界上存在守恒律。 目标是得到存在性、唯一性、稳定性和定性性质。 这些都是目前理论所没有涉及的新型自由边界问题。第一个项目的结果将导致更好地理解裂纹如何在弹性材料中传播，这是一个在电路板布线和老化飞机面板等方面变得越来越重要的问题。 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.