Mathematical Sciences: Regularity for Partial Differential Equations

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9401921 Wang This award supports mathematical research on the regularity theory of partial differential equations and problems of crystal growth. Level surfaces, surfaces where functions are of constant value, are the central subject in studies of phase transition, geometric evolution and free boundary problems. The work follows the spirit of geometric measure theory and incorporates recent work of Caffarelli on free boundaries. Special level surfaces appear as the boundary of crystals and the wave fronts of shock waves in many mathematical models of the physical world. This work seeks to understand the nature, stability and mathematical properties of them. Work will also be done on crystal growth with Gibbs-Thomson effects. Large-time existence of these processes is established using variational methods, which give the most reasonable physical solutions to this long-standing problem. Work will be done on the evolution of crystalline surfaces and crystalline surface energies. The long range goal of the work is to understand the mechanical, chemical, electrical and optical properties associated with solidification. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. Those equations often develop as a result of some minimizing process which takes the mathematical form of a variational problem. The minimizer then must satisfy a corresponding partial differential equation which arises as a solution to this problem. ***