We chose to parametrize the domain D by the conformal map w=g(z) which maps the exterior of the unit circle in the z-plane onto the exterior of D in the w-plane.
According to B.Shraiman and D. Bensimon (Phys. Rev. A30 (1984), 2840-2842), if the conformal map g has only simple poles, then the dynamics of Laplacian growth can be thought of as just the motion of these poles inside the unit circle. They showed that in this case a pole hits the unit circle in finite time, corresponding to a cusp singularity in D.
We looked at the singular case in which the poles (and some branch points) are on the unit circle already. The domain D is a slit domain, and the map g is a Schwarz-Christoffel map.
Again the dynamics of Laplacian growth can be viewed as the motions of the singularities. The gamma's are poles, the beta's are branch points. In addition to simply moving on the unit circle, a pole can split into a pole and two branch points, or two poles and three branch points, corresponding to the development of kinks and "split ends" in the slit domain D.
We came to understand this problem in various ways, but did not advance significantly beyond the discoveries of the 1988 REU group.
The student participants were:
Ed, Alyssa, Sue, and Arti sharing some bright ideas!
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