Abstract of proposed research Ivan Blank
Under this award the investigator will study certain free boundary problems that provide simple models of situations that arise in materials science. One problem is that of finding the optimal placement of a subregion of given volume and different density so that the smallest eigenvalue of an associated operator is minimized. This arises in the study of composite materials and we will study the properties of the interfaces between the two materials for this optimal arrangement. Another problem is that of modeling the behavior of a slow viscous fluid moving between 2 plates and driven by injection through a hole in the upper plate. This is a model for a number of situations in engineering and the diffusion of chemicals in cancerous tumors. Here we are interested in analyzing the moving boundary and its possible changes in topology.
The analysis of these free boundary problems provides information about the interfaces between two materials. It is of importance to describe the possible mathematical phenomenology of these interfaces including their singularities and motion.
