Free Boundary Problems

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We plan to study a variety of free boundary problems. The first one dealswith growing polymeric crystallines. The unknowns are temperature in themelt and the growing surface. The speed of the free boundary is a givenfunction of the temperature, and free boundary points move in such adirection so as to minimize the travel time. We wish to establish theexistence of the solution and to determine the shape of the free boundary.The second problem deals with non-Newtonian jets. We want to approach itvia nonlinear perturbation of the linearized case, a case we have recentlystudied. The next problem is to develop a general bifurcation theory forfree boundary problems. Here we expect to be guided by recent work thatwe have done dealing with special models that arise in mathematicalbiology. In order to determine the stability of the bifurcation, we shallfirst study the asymptotic behavior of solutions of free boundaryproblems, including the Hele-Shaw problem and the evolution of viscousdrops. Subsequently we shall consider the more complicated problems thatarise in mathematical biology, such as the evolution of tumors and ofprotocells. Finally, we shall study the evolution of cracks in elasticmedia. We expect to utilize formulas that we have derived for theevolution of the stress intensity factors.Free boundary problems deal with solving partial differential equations ina domain, a part of whose boundary is unknown in advance; that portion ofthe boundary is called a free boundary. In addition to the usuallyprescribed initial and boundary conditions, an additional condition isimposed at the free boundary, and one seeks to determine both the freeboundary and the solution to the differential equations. A seeminglysmall change in the conditions imposed at the free boundary often resultin, technically, an entirely different problem. Special examples havehistorically guided research in this field; some of the most commonlyknown examples are flow of liquid in contact with air, air streams behindan aircraft, solidification of steel, and melting of solid. Specialexamples motivated by physical models continue to be a driving force inthe development of the field. The present proposal focuses on severaldifferent problems dealing with questions such as crystallization ofpolymers, jet flows for non-Newtonian fluid (e.g. inkjets), bifurcation offree boundary problems arising in biology, nonlinear stability, andpropagation of cracks in elastic media. In all these problems, the goalis to prove that there is a mathematical solution to the scientificproblem and to determine properties of the free boundary.