Mathematical Sciences: Regularity for Partial Differential Equations

9401921王该奖项支持在偏微分方程正则性理论和晶体生长问题方面的数学研究。水平曲面，即函数为常值的曲面，是相变、几何演化和自由边界问题研究的中心课题。这项工作遵循几何测量理论的精神，并结合了卡法雷利最近关于自由边界的工作。在物理世界的许多数学模型中，特殊水平面作为晶体的边界和冲击波的波阵面出现。这项工作旨在了解它们的性质、稳定性和数学性质。还将利用吉布斯-汤姆森效应研究晶体生长。用变分方法确定了这些过程的大时间存在性，为这一长期存在的问题提供了最合理的物理解。工作将在水晶表面和水晶表面能量的演化上完成。这项工作的长期目标是了解与凝固有关的机械、化学、电学和光学性质。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创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。这些方程通常是一些最小化过程的结果，它采用变分问题的数学形式。然后最小化器必须满足相应的偏微分方程，该方程作为该问题的解而出现。＊＊＊

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. ***