We plan to study a variety of free boundary problems. The first one deals with growing polymeric crystallines. The unknowns are temperature in the melt and the growing surface. The speed of the free boundary is a given function of the temperature, and free boundary points move in such a direction so as to minimize the travel time. We wish to establish the existence of the solution and to determine the shape of the free boundary. The second problem deals with non-Newtonian jets. We want to approach it via nonlinear perturbation of the linearized case, a case we have recently studied. The next problem is to develop a general bifurcation theory for free boundary problems. Here we expect to be guided by recent work that we have done dealing with special models that arise in mathematical biology. In order to determine the stability of the bifurcation, we shall first study the asymptotic behavior of solutions of free boundary problems, including the Hele-Shaw problem and the evolution of viscous drops. Subsequently we shall consider the more complicated problems that arise in mathematical biology, such as the evolution of tumors and of protocells. Finally, we shall study the evolution of cracks in elastic media. We expect to utilize formulas that we have derived for the evolution of the stress intensity factors.
Free boundary problems deal with solving partial differential equations in a domain, a part of whose boundary is unknown in advance; that portion of the boundary is called a free boundary. In addition to the usually prescribed initial and boundary conditions, an additional condition is imposed at the free boundary, and one seeks to determine both the free boundary and the solution to the differential equations. A seemingly small change in the conditions imposed at the free boundary often result in, technically, an entirely different problem. Special examples have historically guided research in this field; some of the most commonly known examples are flow of liquid in contact with air, air streams behind an aircraft, solidification of steel, and melting of solid. Special examples motivated by physical models continue to be a driving force in the development of the field. The present proposal focuses on several different problems dealing with questions such as crystallization of polymers, jet flows for non-Newtonian fluid (e.g. inkjets), bifurcation of free boundary problems arising in biology, nonlinear stability, and propagation of cracks in elastic media. In all these problems, the goal is to prove that there is a mathematical solution to the scientific problem and to determine properties of the free boundary.
