Sergio A. Alvarez, Doctor of Philosophy, 1996
University of Maryland, College Park

The Mullins--Sekerka problem is a nonlocal quasilinear free--boundary problem which arises in the study of phase transitions in materials science, as a model of the motion of the phase boundaries or interfaces which separate regions of constant phase during, for example, solidification (W.W. Mullins, R.F. Sekerka. ``Morphological stability of a particle growing by diffusion or heat flow'', Journal of Applied Physics, vol 34, 1963, 323--329). The problem involves the motion of a closed surface inside a bounded domain, driven by its mean curvature and by instantaneous diffusion. We prove local existence, uniqueness, and analytic dependence on the initial data of solutions of the Mullins--Sekerka problem in the case of spatial dimension two, for initial curves which are small perturbations of a disjoint union of circles. The method of maximal regularity (see A. Lunardi, ``Analytic Semigroups and Optimal Regularity in Parabolic Problems'', Birkh\"auser, 1995; S. Angenent, ``Nonlinear Analytic Semiflows'', Proceedings of the Royal Society of Edinburgh, vol 115A, 1990, 91--107) is used in Banach spaces of periodic functions with algebraically decaying Fourier transforms. The analytic core of the proof is the analysis of the integral operators of potential theory in these nonstandard function spaces. The thesis also contains an analysis of the Dirichlet-to-Neumann mapping for multiply connected planar domains in this context. Various results concerning the spaces used in the thesis are given, concerning products, quotients, and analytic images of the associated functions, as well as interpolation properties, and compactness and smoothness of integral operators on these spaces.

Thesis available in DVI.