Mathematical Sciences: Studies in Geometric Analysis and Algebra

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This proposal is for investigation of geometric evolution processes such as might model dendritic growth of a crystal freezing from its melt. The mathematical setting is that of geometric measure theory in which crystals are sets of finite perimeter and temperature and concentration fields are functions belonging to appropriate Sobolev spaces. The final evolution is a limit of discrete time approximations in which crystal geometries are determined by solving variational problems and fields evolve by diffusion equations.