Regularity for Partial Differential Equations

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This project is to study a number of topics about the regularity, and level sets, of solutions of certain elliptic and parabolic partial differential equations. Also some questions about the domains for which the Calderon-Zygmund inequalities hold, parabolic equations are well-posed and the solvability of some degenerate elliptic equations.The degenerate elliptic and parabolic equations to be studied arise as models in geometry, finance and fluid dynamics. The level surfaces are the central subject in the study of phase transitions, geometric evolutions and the free boundary problems. Some special level surfaces appear as the boundary of crystals and the wave fronts of shock waves. The problems to be studied under this project will provide basic information about the solutions of the equations for these various applications.