Mathematical Sciences: Regularity for Partial Differential Equations

小行星9401921 该奖项支持偏微分方程的正则性理论和晶体生长问题的数学研究。 函数为常值的平面是相变、几何演化和自由边界问题研究的中心问题。 这项工作遵循的精神几何措施理论，并结合最近的工作卡法雷利自由边界。 在物理世界的许多数学模型中，特殊的水平面作为晶体的边界和冲击波的波前出现。 这项工作旨在了解它们的性质，稳定性和数学性质。 工作也将在晶体生长与吉布斯-汤姆森效应。 这些过程的大时间存在性是用变分方法建立的，它给出了这个长期存在的问题的最合理的物理解。 工作将在晶体表面和晶体表面能的演变上进行。 这项工作的长期目标是了解与固化相关的机械，化学，电学和光学特性。 偏微分方程是物理世界数学建模的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 这些方程通常是由于某种极小化过程而产生的，它采取了变分问题的数学形式。 最小化，然后必须满足相应的偏微分方程出现作为解决这个问题。 ***

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. ***