Mathematical Sciences: Geometric Measure Theory and the Calculus of Variations

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9626198 Almgren The investigator will continue his long term program of research on equilibrium and dynamic problems in the geometric calculus of variations, largely in the context of geometric measure theory. The investigator plans to exploit his new general isoperimetric inequalities to obtain additional a priori estimates for a family of geometric evolution process problems. In addition, the proposal includes a collaborative project on continuity properties of Steiner symmetrization and other rearrangements on function spaces. A model problem for these studies is creation and analysis of geometric evolution processes including the growth of a crystal or polycrystal in various environments. Possible applications of this research include better ways of controlling the sintering processes used in the manufacture of ceramic materials and coming up with an inventory of computer-aided design examples.