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.